# 2.5E: Limits at Infinity EXERCISES

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## 2.5E: Limits at Infinity EXERCISES

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

251)

$$x=1$$

252)

253)

$$x=−1,x=2$$

254)

255)

$$x=0$$

For the following functions $$f(x)$$, determine whether there is an asymptote at $$x=a$$. Justify your answer without graphing on a calculator.

256) $$f(x)=\frac{x+1}{x^2+5x+4},a=−1$$

257) $$f(x)=\frac{x}{x−2},a=2$$

Yes, there is a vertical asymptote

258) $$f(x)=(x+2)^{3/2},a=−2$$

259) $$f(x)=(x−1)^{−1/3},a=1$$

Yes, there is vertical asymptote

260) $$f(x)=1+x^{−2/5},a=1$$

For the following exercises, evaluate the limit.

261) $$\displaystyle \lim_{x→∞}\frac{1}{3x+6}$$

$$0$$

262) $$\displaystyle \lim_{x→∞}\frac{2x−5}{4x}$$

263) $$\displaystyle \lim_{x→∞}\frac{x^2−2x+5}{x+2}$$

$$∞$$

264) $$\displaystyle \lim_{x→−∞}\frac{3x^3−2x}{x^2+2x+8}$$

265) $$\displaystyle \lim_{x→−∞}\frac{x^4−4x^3+1}{2−2x^2−7x^4}$$

$$−\frac{1}{7}$$

266) $$\displaystyle \lim_{x→∞}\frac{3x}{\sqrt{x^2+1}}$$

267) $$\displaystyle \lim_{x→−∞}\frac{\sqrt{4x^2−1}}{x+2}$$

$$−2$$

268) $$\displaystyle \lim_{x→∞}\frac{4x}{\sqrt{x^2−1}}$$

269) $$\displaystyle \lim_{x→−∞}\frac{4x}{\sqrt{x^2−1}}$$

$$−4$$

270) $$\displaystyle \lim_{x→∞}\frac{2\sqrt{x}}{x−\sqrt{x}+1}$$

For the following exercises, find the horizontal and vertical asymptotes.

271) $$f(x)=x−\frac{9}{x}$$

Horizontal: none, vertical: $$x=0$$

272) $$f(x)=\frac{1}{1−x^2}$$

273) $$f(x)=\frac{x^3}{4−x^2}$$

Horizontal: none, vertical: $$x=±2$$

274) $$f(x)=\frac{x^2+3}{x^2+1}$$

275) $$f(x)=sin(x)sin(2x)$$

Horizontal: none, vertical: none

276) $$f(x)=cosx+cos(3x)+cos(5x)$$

277) $$f(x)=\frac{xsin(x)}{x^2−1}$$

Horizontal: $$y=0,$$ vertical: $$x=±1$$

278) $$f(x)=\frac{x}{sin(x)}$$

279) $$f(x)=\frac{1}{x^3+x^2}$$

Horizontal: $$y=0,$$ vertical: $$x=0$$ and $$x=−1$$

280) $$f(x)=\frac{1}{x−1}−2x$$

281) $$f(x)=\frac{x^3+1}{x^3−1}$$

Horizontal: $$y=1,$$ vertical: $$x=1$$

282) $$f(x)=\frac{sinx+cosx}{sinx−cosx}$$

283) $$f(x)=x−sinx$$

Horizontal: none, vertical: none

284) $$f(x)=\frac{1}{x}−\sqrt{x}$$

For the following exercises, construct a function $$f(x)$$ that has the given asymptotes.

285) $$x=1$$ and $$y=2$$

Answers will vary, for example: $$y=\frac{2x}{x−1}$$

286) $$x=1$$ and $$y=0$$

287) $$y=4, x=−1$$

Answers will vary, for example: $$y=\frac{4x}{x+1}$$

288) $$x=0$$

CHAPTER REVIEW EXERCISES

CR 1) $$\displaystyle \lim_{x→∞}\frac{3x\sqrt{x^2+1}}{\sqrt{x^4−1}}$$

$$3$$

CR 2) $$\displaystyle \lim_{x→∞}cos(\frac{1}{x})$$

CR 3) $$\displaystyle \lim_{x→1}\frac{x−1}{sin(πx)}$$

$$−\frac{1}{π}$$
CR 4) $$\displaystyle \lim_{x→∞}(3x)^{1/x}$$