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  • Evaluate <img alt="\mathrm{ \lim _{x \rightarrow \infty} \frac{\cos x+\sin x}{x^2}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Clim%20_%7Bx%20%5Crightarrow%20%5Cinf

Evaluate \mathrm{ \lim _{x \rightarrow \infty} \frac{\cos x+\sin x}{x^2}}

Option: 1

0


Option: 2

2


Option: 3

-1


Option: 4

1


Answers (1)

best_answer

You can't use Taylor series in this case, in fact the approximations \mathrm{\sin x \sim x} and \mathrm{\cos x \sim 1-\frac{x^2}{2}} work only when \mathrm{x \rightarrow 0 } but in this limit x approaches infinity. There's no need to use sophisticated tools, just use the squeeze theorem:

\mathrm{\cos x \sim0 \leq \frac{|\sin x+\cos x|}{x^2} \leq \frac{2}{x^2} \text { and } \frac{2}{x^2} \rightarrow 0}

Hence you have:

\mathrm{\lim _{x \rightarrow+\infty} \frac{\sin x+\cos x}{x^2}=0}
 

Posted by

SANGALDEEP SINGH

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