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# 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.

J46) $$\displaystyle \lim_{x→10}f(x)=0$$

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

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

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

J49) $$\displaystyle \lim_{x→6}f(x)=5$$

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

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

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

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) $$\displaystyle \lim_{x→1}\frac{x^3−1}{x^2−1}$$

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

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

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

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

State the vertical asymptote for each function, if any.

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

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

$$x=4$$

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

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

$$x=0$$
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 $$x=\frac{\pi}{2}$$?