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

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

2


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-x^{2} / 2} work only when \mathrm{x \rightarrow 0} but in this limit \mathrm{x } approaches infinity. There's no need to use sophisticated tools, just use the squeeze theorem:

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

Hence you have:

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

Posted by

Deependra Verma

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