Math  /  Calculus

QuestionFor the following exercises, find the horizontal and vertical asymptotes.
271. f(x)=x9xf(x)=x-\frac{9}{x}
272. f(x)=11x2f(x)=\frac{1}{1-x^{2}}
273. f(x)=x34x2f(x)=\frac{x^{3}}{4-x^{2}}
274. f(x)=x2+3x2+1f(x)=\frac{x^{2}+3}{x^{2}+1}
275. f(x)=sin(x)sin(2x)f(x)=\sin (x) \sin (2 x)
276. f(x)=cosx+cos(3x)+cos(5x)f(x)=\cos x+\cos (3 x)+\cos (5 x)
277. f(x)=xsin(x)x21f(x)=\frac{x \sin (x)}{x^{2}-1}
278. f(x)=xsin(x)f(x)=\frac{x}{\sin (x)}
279. f(x)=1x3+x2f(x)=\frac{1}{x^{3}+x^{2}}
280. f(x)=1x12xf(x)=\frac{1}{x-1}-2 x

Studdy Solution
Analyze f(x)=1x12x f(x) = \frac{1}{x-1} - 2x :
- Horizontal Asymptote: As x± x \to \pm \infty , f(x)2x f(x) \to -2x , so no horizontal asymptote. - Vertical Asymptote: At x=1 x = 1 , since the denominator of the first term is zero.

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