Can anyone show me step by step how to determine the following limit?
$\lim _{x\to \infty \:}\left(\frac{\sin \left(\sin \left(x\right)\right)}{x}\right)$
Can anyone show me step by step how to determine the following limit?
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limits
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0Hint: if only there was a $\sin(x)$ on the bottom... – 2017-01-19
2 Answers
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since $$ |\sin(\cdot)|\le 1 $$ then using the above inequality you can show that: $$ 0\le\left|\frac{\sin \left(\sin \left(x\right)\right)}{x}\right|\le\frac{1}{|x|} \to 0 \,\,\,{\rm{for}}\,\,\,x\to\infty $$ by the squeeze theorem the limit is $0$.
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$| \sin(\sin(x))| \le 1$ for all $x$. Hence the limit in question is $=0$