Limits & Continuity

Problem 1

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.

See Solution

Problem 2

Find the limit: $\lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1}$ . Options: (a) $\frac{2}{e}$ (b) 1 (c) 0 (d) nonexistent.

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Problem 3

8) Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left((x+h)^{2}\right)-\cos \left(x^{2}\right)}{h}$ .
9) Determine $x$ for the maximum of $y=\frac{4}{3} x^{3}-8 x^{2}+15 x$ on $[0,4]$ .

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Problem 4

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}$ . Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

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Problem 5

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . What is it?

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Problem 6

Find the limit of $f(x)=\frac{\sin 4x}{4 \sin 3x}$ as $x$ approaches 0.

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Problem 7

Find the limit of $f(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}$ as $x \to 0$ .

See Solution

Problem 8

Find the limit: $\lim_{x \rightarrow 0} f(x)$ where $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ .

See Solution

Problem 9

Find the limit: If $f(z)=\frac{z+\bar{z}}{z}$ , then $\lim _{x \rightarrow 0} f(x+0 i)=$ .

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Problem 10

Find $\lim _{x \rightarrow 0} f(x+0 i)$ and $\lim _{y \rightarrow 0} f(0+i y)$ for $f(z)=\frac{z+\bar{z}}{z}$ .

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Problem 11

Evaluate the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}$ .

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Problem 12

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

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Problem 13

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

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Problem 14

Find the slant asymptotes of the curve given by the equation $y=\sqrt{x^{2}+4 x}$ .

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Problem 15

Evaluate the following limits as $x$ approaches 0:
1. $\lim _{x \rightarrow 0} \frac{(1+x) \sin x-x \cos x}{x^{2}}$
2. $\lim _{x \rightarrow 0} \frac{e^{x^{2}}-\cos x}{x \sin x}$
3. $\lim _{x \rightarrow 0} \frac{\sin x-x e^{x}+x^{2}}{x(\cos x-1)}$
4. $\lim _{x \rightarrow 0}\left\{\frac{1}{\sin ^{2} x}-\frac{1}{x^{2}}\right\}$

See Solution

Problem 16

Calculate the average rate of change of $f(x)=2 x^{4}-3 x$ from $x=-1$ to $x=2$ .

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Problem 17

Check if the function $f(x)$ is continuous at $x=-3$ where $f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}$ .

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Problem 18

Find the limit as $x$ approaches 0 for $\frac{6(1-\cos x)}{x^{2}}$ .

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Problem 19

Find the limit: $\operatorname{Lim}_{x \rightarrow-3} \frac{4 x^{2}+9 x-9}{x^{3}+27}$ .

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Problem 20

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{-1 / x^{9}}}{x^{2}}$

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Problem 21

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^{2}}}}{x^{2}}$ .

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Problem 22

Find the limit: $A=\lim _{x \rightarrow+\infty} \frac{2x^{2}-2x-1}{x^{2}+1}$ .

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Problem 23

Simplify the expression: $\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}$ .

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Problem 24

Find the limit as $x$ approaches infinity: $x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}$ .

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Problem 25

Find the limit as $n \to \infty$ of $\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}$ .

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Problem 26

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

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Problem 27

곡선 $y=\frac{4}{x}$ 위의 점 A(1,4)와 B(t, $\frac{4}{t}$ )를 지나는 직선의 삼각형 OPB 넓이 $S(t)$ 의 $\lim _{t \rightarrow \infty} S(t)$ 값은?

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Problem 28

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Options: (A) $\frac{1}{2 \pi}$ (B) $\frac{1}{\pi}$ (C) 1 (D) nonexistent.

See Solution

Problem 29

Find $\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)$ .

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Problem 30

Find the limit as $x$ approaches 0 for $\cos \left(\frac{1+\pi x}{x}\right)$ .

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Problem 31

Find the limit as $x$ approaches 4 for the piecewise function $f(x)$ defined as: $f(x) = -7x + 46$ if $x < 4$ , $12$ if $x = 4$ , $2x^2 - 14$ if $x > 4$ . What is $\lim_{x \rightarrow 4} f(x)$ ?

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Problem 32

Find the average velocity of a ball thrown with $y=41t-22t^{2}$ at $t=2$ for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

See Solution

Problem 33

Calculate the average rate of change of the function $k(x)=-16 \sqrt{x}$ from $x=12$ to $x=15$ .

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Problem 34

Does the function $f(x)=x$ have a limit as $x$ approaches 3 from all real numbers except 3?

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Problem 35

Find $\lim _{x \rightarrow 2} f(x)$ given that $\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9$ and $f(x) \geq 0$ .

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Problem 36

Find the limit as $x$ approaches 3 for the expression $x^{-1/2}(5x-7)^{1/3}$ .

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Problem 37

Find the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

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Problem 38

Find the limit: $\lim _{x \rightarrow 0} \frac{(3+x)^{2}-9}{x}$ .

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Problem 39

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

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Problem 40

Find the limit as $x$ approaches 0 for the expression $\frac{\sin x}{x}$ .

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Problem 41

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

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Problem 42

Find the limit as $x$ approaches -1 for the expression $-2x^3 - x + 5$ .

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Problem 43

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}-x-2}{x+1}$ .

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Problem 44

Find the limit: $\lim _{x \rightarrow 4} \sqrt{1+2 x}$ .

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Problem 45

Find the limit as $x$ approaches 0 for the expression $e^{-x} \cos x$ .

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Problem 46

Find the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+1}-1}{x}$

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Problem 47

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$ .

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Problem 48

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3}{x^{2}}$ , where $h \neq 0$ .

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Problem 49

Find the limit: $\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right)$ .

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Problem 50

Find the limit as $x$ approaches $\frac{11}{10}$ from the right: $\lim _{x \rightarrow \frac{11}{10}^{+}}\left(\frac{15 x}{11-10 x}\right)$ .

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Problem 51

Find the limits: 1) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x}$ 2) $\lim _{x \rightarrow 3} \frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

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Problem 52

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}-9}{x^{2}-3x}$ .

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Problem 53

Simplify the expression: $\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

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Problem 54

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function: $f(x) = -9 + 4x$ (for $x<3$ ) and $f(x) = -6 + x^{2}$ (for $x>3$ ).

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Problem 55

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function $f(x)=\begin{cases} x^{2}-7 & x>3 \\ -4+2x & x<3 \end{cases}$ .

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Problem 56

Find the limit: $\lim _{x \rightarrow 0^{+}} x^{x}$ .

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Problem 57

Sketch the graph of the piecewise function and find the limits:
1. For $f(x) = \begin{cases} \sin x & x < 0 \\ x^2 & 0 \leq x < 2 \\ x & x \geq 2 \end{cases}$ , find: i. $\lim_{x \to 0} f(x)$ ii. $\lim_{x \to 2} f(x)$
2. For $f(x) = \begin{cases} e^x & x \leq 0 \\ |x| + 1 & 0 < x < 1 \\ \ln x & x \geq 1 \end{cases}$ , find: i. $\lim_{x \to 0} f(x)$ ii. $\lim_{x \to 1} f(x)$

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Problem 58

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-2 x-24}{6 x^{2}+23 x-4}$ .

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Problem 59

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{x+4}{2-3 x}\right)^{3}$ .

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Problem 60

Find the limit as $x$ approaches infinity of $\left(\frac{x+4}{2-3 x}\right)^{3}$ .

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Problem 61

Find the limit: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}$ .

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Problem 62

Find the limit as $x$ approaches -3 for the expression $\frac{x^{2}+9x+18}{x+3}$ .

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Problem 63

Find the limit: $\lim _{x \rightarrow 0} \frac{-3 x}{\sqrt{3 x+64}-8}$ .

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Problem 64

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

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Problem 65

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[8,8+h]$ . Express your answer in terms of $h$ .

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Problem 66

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

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Problem 67

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

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Problem 68

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

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Problem 69

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

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Problem 70

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

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Problem 71

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

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Problem 72

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

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Problem 73

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

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Problem 74

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .

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Problem 75

Evaluate $f(x)=\frac{x^{2}-8 x}{x^{2}-7 x-8}$ at $x=0,-0.5,-0.9,-0.95,-0.99,-0.999,-2,-1.5,-1.1,-1.01,-1.001$ . Find $\lim_{x \to -1} f(x)$ to six decimal places.

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Problem 76

Find $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

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Problem 77

Find the limit: $\lim _{x \rightarrow 2}[f(x)+5 g(x)]$ given $\lim _{x \rightarrow 2} f(x)=9$ and $\lim _{x \rightarrow 2} g(x)=-5$ .

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Problem 78

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+3 x}{x^{2}-5 x-24}$ (Enter DNE if it doesn't exist.)

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Problem 79

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+6 x+4}{x-2}$ . If it doesn't exist, enter DNE.

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Problem 80

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ . Simplify and evaluate the limit, or state DNE.

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Problem 81

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ and simplify the expression.

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Problem 82

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ . Simplify and evaluate if possible.

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Problem 83

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(h-4)^{2}-16}{h}$ . If it doesn't exist, write DNE.

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Problem 84

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify the expression.

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Problem 85

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify to $\lim _{x \rightarrow 2}(\sqrt{x+2}+2 x)$ .

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Problem 86

Evaluate the limits and value for the piecewise function $g(x)$ at $x=1$ and $x=2$ .

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Problem 87

Find the limit as $t$ approaches 0 from the right of $\log(t)$ .

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Problem 88

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for the function $f(x)=\sqrt{x}$ .

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Problem 89

Find the limit: $\lim _{t \rightarrow 0^{-}} \ln (-t)$

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Problem 90

Calculate the limit: $\lim _{t \rightarrow 5^{+}} \ln (t-5)$ .

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Problem 91

Find the largest $\delta$ for $\varepsilon=0.2$ and $\varepsilon=0.1$ in the limit $\lim _{x \rightarrow 2}(x^{3}-5 x+5)=3$ . Round to four decimal places.

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Problem 92

Find $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6}-3 x-4=-22$ .

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Problem 93

Prove that $\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}$ by finding $\delta$ in terms of $\varepsilon$ .

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Problem 94

Find $\delta$ for $f(x)=x^{\frac{2}{3}}$ as $x \to 1$ with $\varepsilon=0.001$ such that $|f(x)-L|<\varepsilon$ .

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Problem 95

Prove that $\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}$ by finding $\delta$ for any $\varepsilon>0$ .

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Problem 96

Find the largest $\delta$ for $\lim _{x \rightarrow 2}\left(x^{3}-5 x+5\right)=3$ with $\varepsilon=0.2$ and $\varepsilon=0.1$ . Round to four decimal places.

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Problem 97

Find the limit: $\lim _{x \rightarrow 2}\left(\frac{9 x-6}{3 x^{2}-10 x+8}-\frac{9 x+24}{4 x^{2}-9 x+2}\right)$ .

See Solution

Problem 98

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-6x+5$ , with $h \neq 0$ . Simplify your answer.

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Problem 99

Given the function $f(x)=\left\{\begin{array}{lll}2 x+18 & \text { if } & x<-6 \\ \sqrt{x+42} & \text { if } & x>-6 \\ 2 & \text { if } & x=-6\end{array}\right.$ , determine the truth of the following statements about $f(-6)$ and its limits.

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Problem 100

Find the limit of the fish length model $L(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t}$ as $t \to \infty$ .

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Limits & Continuity

Problem 1

Find the limit of the sequence an=n2+3n−na_{n}=\sqrt{n^{2}+3 n}-nan​=n2+3n​−n. Options: A) 3 B) 2 C) 1/2 1 / 2 1/2 D) 3/2 3 / 2 3/2 E) 0 F) 1 G) ∞ \infty ∞ H) does not exist.

Problem 2

Find the limit: lim⁡x→12⋅ln⁡(x)ex−1 \lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1} limx→1​ex−12⋅ln(x)​. Options: (a) 2e \frac{2}{e} e2​ (b) 1 (c) 0 (d) nonexistent.

Problem 3

Problem 4

Find the limit: lim⁡x→∞4⋅ln⁡(x)+43x\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}x→∞lim​3x4⋅ln(x)+4​. Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

Problem 5

Find the limit: lim⁡x→πcos⁡x+sin⁡(2x)+1x2−π2\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}limx→π​x2−π2cosx+sin(2x)+1​. What is it?

Problem 6

Find the limit of f(x)=sin⁡4x4sin⁡3xf(x)=\frac{\sin 4x}{4 \sin 3x}f(x)=4sin3xsin4x​ as xxx approaches 0.

Problem 7

Find the limit of f(x)=sin⁡4x4sin⁡3x+sin⁡(x/3)9xf(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}f(x)=4sin3xsin4x​+9xsin(x/3)​ as x→0x \to 0x→0.

Problem 8

Find the limit: lim⁡x→0f(x)\lim_{x \rightarrow 0} f(x)limx→0​f(x) where f(x)=2xcos⁡(1x2)f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)f(x)=2xcos(x21​).

Problem 9

Find the limit: If f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z}f(z)=zz+zˉ​, then lim⁡x→0f(x+0i)=\lim _{x \rightarrow 0} f(x+0 i)=limx→0​f(x+0i)=.

Problem 10

Find lim⁡x→0f(x+0i)\lim _{x \rightarrow 0} f(x+0 i)limx→0​f(x+0i) and lim⁡y→0f(0+iy)\lim _{y \rightarrow 0} f(0+i y)limy→0​f(0+iy) for f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z}f(z)=zz+zˉ​.

Problem 11

Evaluate the limit as xxx approaches 0: lim⁡x→0x2−25x2−4x−5\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}limx→0​x2−4x−5x2−25​.

Problem 12

Find discontinuities of f(x)={x2−1,x<1x,x≥1f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.f(x)={x2−1,x,​x<1x≥1​ and explain why.

Problem 13

Show that the limit limit⁡x→2∣x−2∣x−2\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}limitx→2​x−2∣x−2∣​ does not exist.

Problem 14

Find the slant asymptotes of the curve given by the equation y=x2+4xy=\sqrt{x^{2}+4 x}y=x2+4x​.

Problem 15

Problem 16

Calculate the average rate of change of f(x)=2x4−3xf(x)=2 x^{4}-3 xf(x)=2x4−3x from x=−1x=-1x=−1 to x=2x=2x=2.

Problem 17

Check if the function f(x)f(x)f(x) is continuous at x=−3x=-3x=−3 where f(x)={6+x2,x≥−34−3x,x<−3f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}f(x)={6+x2,4−3x,​x≥−3x<−3​.

Problem 18

Find the limit as xxx approaches 0 for 6(1−cos⁡x)x2\frac{6(1-\cos x)}{x^{2}}x26(1−cosx)​.

Problem 19

Find the limit: Lim⁡x→−34x2+9x−9x3+27\operatorname{Lim}_{x \rightarrow-3} \frac{4 x^{2}+9 x-9}{x^{3}+27}Limx→−3​x3+274x2+9x−9​.

Problem 20

Evaluate the limit: lim⁡x→0e−1/x9x2\lim _{x \rightarrow 0} \frac{e^{-1 / x^{9}}}{x^{2}}x→0lim​x2e−1/x9​

Problem 21

Find the limit: lim⁡x→0e−1x2x2\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^{2}}}}{x^{2}}limx→0​x2e−x21​​.

Problem 22

Find the limit: A=lim⁡x→+∞2x2−2x−1x2+1A=\lim _{x \rightarrow+\infty} \frac{2x^{2}-2x-1}{x^{2}+1}A=limx→+∞​x2+12x2−2x−1​.

Problem 23

Simplify the expression: (x+Δx)2−9(x+Δx)+8−(x2−9x+8)Δx\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}Δx(x+Δx)2−9(x+Δx)+8−(x2−9x+8)​.

Problem 24

Find the limit as xxx approaches infinity: x{tan⁡−1[(x+1)(x−2)]−π4}x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}x{tan−1[(x−2)(x+1)​]−4π​}.

Problem 25

Find the limit as n→∞n \to \inftyn→∞ of ⌊x⌋+⌊22x⌋+…+⌊n2x⌋n3\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}n3⌊x⌋+⌊22x⌋+…+⌊n2x⌋​.

Problem 26

Find the limit: lim⁡n→∞(1+x)(1+x2)(1+x4)⋯(1+x2n)\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})limn→∞​(1+x)(1+x2)(1+x4)⋯(1+x2n) for ∣x∣<1|x|<1∣x∣<1.

Problem 27

곡선 y=4xy=\frac{4}{x}y=x4​ 위의 점 A(1,4)와 B(t, 4t\frac{4}{t}t4​)를 지나는 직선의 삼각형 OPB 넓이 S(t)S(t)S(t)의 lim⁡t→∞S(t)\lim _{t \rightarrow \infty} S(t)limt→∞​S(t) 값은?

Problem 28

Find the limit: lim⁡x→πcos⁡x+sin⁡(2x)+1x2−π2\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}limx→π​x2−π2cosx+sin(2x)+1​. Options: (A) 12π\frac{1}{2 \pi}2π1​ (B) 1π\frac{1}{\pi}π1​ (C) 1 (D) nonexistent.

Problem 29

Find lim⁡x→∞cos⁡(1+πxx)\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)limx→∞​cos(x1+πx​).

Problem 30

Find the limit as xxx approaches 0 for cos⁡(1+πxx)\cos \left(\frac{1+\pi x}{x}\right)cos(x1+πx​).

Problem 31

Find the limit as xxx approaches 4 for the piecewise function f(x)f(x)f(x) defined as: f(x)=−7x+46f(x) = -7x + 46f(x)=−7x+46 if x<4x < 4x<4, 121212 if x=4x = 4x=4, 2x2−142x^2 - 142x2−14 if x>4x > 4x>4. What is lim⁡x→4f(x)\lim_{x \rightarrow 4} f(x)limx→4​f(x)?

Problem 32

Find the average velocity of a ball thrown with y=41t−22t2y=41t-22t^{2}y=41t−22t2 at t=2t=2t=2 for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

Problem 33

Calculate the average rate of change of the function k(x)=−16xk(x)=-16 \sqrt{x}k(x)=−16x​ from x=12x=12x=12 to x=15x=15x=15.

Problem 34

Does the function f(x)=xf(x)=xf(x)=x have a limit as xxx approaches 3 from all real numbers except 3?

Problem 35

Find lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) given that lim⁡x→2[f(x)]2−8x+3x+1=9\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9limx→2​x+1[f(x)]2−8x+3​​=9 and f(x)≥0f(x) \geq 0f(x)≥0.

Problem 36

Find the limit as xxx approaches 3 for the expression x−1/2(5x−7)1/3x^{-1/2}(5x-7)^{1/3}x−1/2(5x−7)1/3.

Problem 37

Find the limit: lim⁡x→3x2−x−6x−3\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}limx→3​x−3x2−x−6​.

Problem 38

Find the limit: lim⁡x→0(3+x)2−9x\lim _{x \rightarrow 0} \frac{(3+x)^{2}-9}{x}limx→0​x(3+x)2−9​.

Problem 39

Find the limit: lim⁡x→04+x−2x\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}limx→0​x4+x​−2​.

Problem 40

Find the limit as xxx approaches 0 for the expression sin⁡xx\frac{\sin x}{x}xsinx​.

Problem 41

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.

Find the limit: $\lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1}$ . Options: (a) $\frac{2}{e}$ (b) 1 (c) 0 (d) nonexistent.

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}$ . Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . What is it?

Find the limit of $f(x)=\frac{\sin 4x}{4 \sin 3x}$ as $x$ approaches 0.

Find the limit of $f(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}$ as $x \to 0$ .

Find the limit: $\lim_{x \rightarrow 0} f(x)$ where $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ .

Find the limit: If $f(z)=\frac{z+\bar{z}}{z}$ , then $\lim _{x \rightarrow 0} f(x+0 i)=$ .

Find $\lim _{x \rightarrow 0} f(x+0 i)$ and $\lim _{y \rightarrow 0} f(0+i y)$ for $f(z)=\frac{z+\bar{z}}{z}$ .

Evaluate the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}$ .

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

Find the slant asymptotes of the curve given by the equation $y=\sqrt{x^{2}+4 x}$ .

Calculate the average rate of change of $f(x)=2 x^{4}-3 x$ from $x=-1$ to $x=2$ .

Check if the function $f(x)$ is continuous at $x=-3$ where $f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}$ .

Find the limit as $x$ approaches 0 for $\frac{6(1-\cos x)}{x^{2}}$ .

Find the limit: $\operatorname{Lim}_{x \rightarrow-3} \frac{4 x^{2}+9 x-9}{x^{3}+27}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{-1 / x^{9}}}{x^{2}}$

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^{2}}}}{x^{2}}$ .

Find the limit: $A=\lim _{x \rightarrow+\infty} \frac{2x^{2}-2x-1}{x^{2}+1}$ .

Simplify the expression: $\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}$ .

Find the limit as $x$ approaches infinity: $x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}$ .

Find the limit as $n \to \infty$ of $\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}$ .

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

곡선 $y=\frac{4}{x}$ 위의 점 A(1,4)와 B(t, $\frac{4}{t}$ )를 지나는 직선의 삼각형 OPB 넓이 $S(t)$ 의 $\lim _{t \rightarrow \infty} S(t)$ 값은?

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Options: (A) $\frac{1}{2 \pi}$ (B) $\frac{1}{\pi}$ (C) 1 (D) nonexistent.

Find $\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)$ .

Find the limit as $x$ approaches 0 for $\cos \left(\frac{1+\pi x}{x}\right)$ .

Find the limit as $x$ approaches 4 for the piecewise function $f(x)$ defined as: $f(x) = -7x + 46$ if $x < 4$ , $12$ if $x = 4$ , $2x^2 - 14$ if $x > 4$ . What is $\lim_{x \rightarrow 4} f(x)$ ?

Find the average velocity of a ball thrown with $y=41t-22t^{2}$ at $t=2$ for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

Calculate the average rate of change of the function $k(x)=-16 \sqrt{x}$ from $x=12$ to $x=15$ .

Does the function $f(x)=x$ have a limit as $x$ approaches 3 from all real numbers except 3?

Find $\lim _{x \rightarrow 2} f(x)$ given that $\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9$ and $f(x) \geq 0$ .

Find the limit as $x$ approaches 3 for the expression $x^{-1/2}(5x-7)^{1/3}$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{(3+x)^{2}-9}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

Find the limit as $x$ approaches 0 for the expression $\frac{\sin x}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

Find the limit as $x$ approaches -1 for the expression $-2x^3 - x + 5$ .

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}-x-2}{x+1}$ .

Find the limit: $\lim _{x \rightarrow 4} \sqrt{1+2 x}$ .

Find the limit as $x$ approaches 0 for the expression $e^{-x} \cos x$ .

Find the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+1}-1}{x}$

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3}{x^{2}}$ , where $h \neq 0$ .

Find the limit: $\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right)$ .

Find the limit as $x$ approaches $\frac{11}{10}$ from the right: $\lim _{x \rightarrow \frac{11}{10}^{+}}\left(\frac{15 x}{11-10 x}\right)$ .

Find the limits: 1) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x}$ 2) $\lim _{x \rightarrow 3} \frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}-9}{x^{2}-3x}$ .

Simplify the expression: $\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function: $f(x) = -9 + 4x$ (for $x<3$ ) and $f(x) = -6 + x^{2}$ (for $x>3$ ).

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function $f(x)=\begin{cases} x^{2}-7 & x>3 \\ -4+2x & x<3 \end{cases}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}} x^{x}$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-2 x-24}{6 x^{2}+23 x-4}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{x+4}{2-3 x}\right)^{3}$ .

Find the limit as $x$ approaches infinity of $\left(\frac{x+4}{2-3 x}\right)^{3}$ .

Find the limit: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}$ .

Find the limit as $x$ approaches -3 for the expression $\frac{x^{2}+9x+18}{x+3}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{-3 x}{\sqrt{3 x+64}-8}$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[8,8+h]$ . Express your answer in terms of $h$ .

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .