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Mathematics LibreTexts

2.4E: Infinite Limits EXERCISES

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2.4: Infinite Limit Exercises

In the following exercises, find the limit.

In the following exercises, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false.

CNX_Calc_Figure_02_02_201.jpeg

J46) \displaystyle \lim_{x→10}f(x)=0

J47) \displaystyle \lim_{x→−2^+}f(x)=3

Answer:

False; \displaystyle \lim_{x→−2^+}f(x)=+∞

J48) \displaystyle \lim_{x→−8}f(x)=f(−8)

J49) \displaystyle \lim_{x→6}f(x)=5

Answer:

False; \displaystyle \lim_{x→6}f(x) DNE since\displaystyle \lim_{x→6^−}f(x)=2 and \displaystyle \lim_{x→6^+}f(x)=5.

J2.4.1)

a. \displaystyle \lim_{x→−3^+}\frac{x}{x+3}

b. \displaystyle \lim_{x→−3^-}\frac{x}{x+3}

c. \displaystyle \lim_{x→−3}\frac{x}{x+3}


Answer:
a. −∞
b. ∞
c. DNE

J2.4.2) \displaystyle \lim_{x→0}\ln |x|

J2.4.3)

a. \displaystyle \lim_{x→5^+}\frac{2}{x-5}

b. \displaystyle \lim_{x→5^-}\frac{2}{x-5}

c. \displaystyle \lim_{x→5}\frac{2}{x-5}

Answer:
a. ∞
b. −∞
c. DNE

J2.4.4)

a. \displaystyle \lim_{x→-2^+}\frac{x}{(x+2)^2}

b. \displaystyle \lim_{x→-2^-}\frac{x}{(x+2)^2}

c. \displaystyle \lim_{x→-2}\frac{x}{(x+2)^2}

J2.4.5)

a. \displaystyle \lim_{x→6^+}\frac{x}{(6-x)^2}

b. \displaystyle \lim_{x→6^-}\frac{x}{(6-x)^2}

c. \displaystyle \lim_{x→6}\frac{x}{(6-x)^2}

Answer:
a. ∞
b. ∞
c. ∞

J2.4.6)

a. \displaystyle \lim_{x→1^+}\frac{2x^2+7x−4}{x^2+x−2}

b. \displaystyle \lim_{x→1^−}\frac{2x^2+7x−4}{x^2+x−2}

c.\displaystyle \lim_{x→1}\frac{2x^2+7x−4}{x^2+x−2}

J2.4.7) \displaystyle \lim_{x→1}\frac{x^3−1}{x^2−1}

Answer:
\displaystyle lim_{x→1}\frac{x^3−1}{x^2−1}=\displaystyle \lim_{x→1}\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)}=\displaystyle \lim_{x→1}\frac{x^2+x+1}{x+1}=\frac{3}{2}

J2.4.8) \displaystyle \lim_{x→1/2}\frac{2x}{2x−1}

J2.4.9) \displaystyle \lim_{x→1/2}\frac{2x^2+3x−2}{2x−1}

Answer:
\displaystyle \lim_{x→ 1/2}\frac{2x^2+3x−2}{2x−1}=\displaystyle \lim_{x→1/2}\frac{(2x−1)(x+2)}{2x−1}=\displaystyle \lim_{x→1/2}(x+2)=\frac{5}{2}

State the vertical asymptote for each function, if any.

J2.4.10) f(x)=\ln x

J2.4.11) g(x)=\frac{x+5}{x-4}

Answer:
x=4

J2.4.12) g(x)=\frac{7}{x+5}

J2.4.13) g(x)=\frac{7}{x}

Answer:
x=0

J2.4.14)

a. \displaystyle \lim_{x→\frac{\pi}{2}^+}\tan x=

b. \displaystyle \lim_{x→\frac{\pi}{2}^-}\tan x=

c. \displaystyle \lim_{x→\frac{\pi}{2}}\tan x=

d. Does f(x)=\tan x have a vertical asymptote at x=\frac{\pi}{2}?


2.4E: Infinite Limits EXERCISES is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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