Find the value of lim _{x rightarrow 0} frac{ | vedclass

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(C) We are given the limit $L = \lim _{x \rightarrow 0} \frac{\sin (x^m)}{(\sin x)^n}$ where $n < m$. By multiplying and dividing by $x^m$ in the nume

Vedclass · Accepted Answer

$0$

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(C) We are given the limit $L = \lim _{x \rightarrow 0} \frac{\sin (x^m)}{(\sin x)^n}$ where $n By multiplying and dividing by $x^m$ in the numerator and $x^n$ in the denominator,we get: $L = \lim _{x \rightarrow 0} \left( \frac{\sin (x^m)}{x^m} \cdot x^m \right) / \left( \frac{\sin x}{x} \cdot x \right)^n$ $L = \lim _{x \rightarrow 0} \left( \frac{\sin (x^m)}{x^m} \right) \cdot \left( \frac{x}{\sin x} \right)^n \cdot x^{m-n}$ Since $\lim _{x \rightarrow 0} \frac{\sin (x^m)}{x^m} = 1$ and $\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1$,the expression becomes: $L = 1 \cdot (1)^n \cdot \lim _{x \rightarrow 0} x^{m-n}$ Given $n  0$. Therefore,$L = 0^{m-n} = 0$. Hence,option $C$ is correct.

Find the value of $\lim _{x \rightarrow 0} \frac{\sin (x^m)}{(\sin x)^n}$,given that $n < m$.

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