Evaluate the following limit: _ x 0 5x 3x - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}}{\tan3\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}}{\tan3\text{x}}$ $=\frac{\lim\limits_{\text{x}\rightarrow0}{\sin5\text{x}}{}}{\lim\limits_{3\text{x}\rightarrow0}{\tan3\text{x}}}$ $=\frac{\lim\limits_{5\text{x}\rightarrow0}\frac{\sin5\text{x}}{5\text{x}}\times5}{\lim\limits_{3\text{x}\rightarrow0}\frac{\tan3\text{x}}{3\text{x}}\times3}$ $[\because$ if x → 0 then 3x → 0 also 5x → 0$]$ $=\frac{5}{3}\times1$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1,\text{ also }\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$ $=\frac{5}{3}$

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