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Q.
If $m , n \in N , \displaystyle\lim _{ x \rightarrow 0} \frac{\sin x ^{ n }}{(\sin x )^{ m }}$ is
Limits and Derivatives
Solution:
$L=\displaystyle\lim _{x \rightarrow 0} \frac{\sin x^{ n }}{(\sin x)^{ m }}=\displaystyle\lim _{x \rightarrow 0} \frac{\frac{\sin x^{ n }}{x^{ n }} x^{ n }}{\frac{(\sin x)^{ m }}{x^{ m }} x^{ m }}=\displaystyle\lim _{x \rightarrow 0} x^{ n - m }$
If $n = m$, then
$L =(\text { a very small value near to zero })^{\text {exactly zero }}=1$
If $n > m$, then
$L =(\text { a very small value near to zero })^{\text {positive integer }}=0$
If $n < m$, then
$L =(\text { a very small value near to zero })^{\text {negative integer }}=\infty$