# Evaluate the following limits. lim_{xrightarrow0}frac{tan5x}{x}

Question
Limits and continuity
Evaluate the following limits.
$$\lim_{x\rightarrow0}\frac{\tan5x}{x}$$

2021-02-23
Given,
$$\lim_{x\rightarrow0}\frac{\tan5x}{x}$$
On simplification, we get
$$\lim_{x\rightarrow0}\frac{\tan5x}{x}=\lim_{x\rightarrow0}\frac{5\tan5x}{5x}$$
$$=5\cdot\lim_{x\rightarrow0}\frac{\tan5x}{5x}$$
$$=5\cdot\lim_{5x\rightarrow0}\frac{\tan5x}{5x}\ \ [\because As\ x\rightarrow0,5x\rightarrow0]$$
$$=5\cdot(1)\ \ [\because\lim_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1]$$
$$=5$$

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