Evaluate the following limit: _ x 0 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{m}\text{x}}{\tan\text{n}\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{m}\text{x}}{\tan\text{n}\text{x}}$$=\frac{\lim\limits_{\text{x}\rightarrow0}{\tan\text{m}\text{x}}{}}{\lim\limits_{\text{x}\rightarrow0}{\sin\text{n}\text{x}}}$
$=\frac{\lim\limits_{\text{m}\text{x}\rightarrow0}\frac{\tan\text{m}\text{x}}{\text{m}\text{x}}\times\text{m}\text{x}}{\lim\limits_{\text{n}\text{x}\rightarrow0}\frac{\tan\text{n}\text{x}}{\text{n}\text{x}}\times\text{n}\text{x}}$ $[\because$ if x → 0 then mx → 0 also nx → 0$]$
$=\frac{1\times\text{m}}{1\times\text{n}}$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$
$=\frac{\text{m}}{\text{n}}$

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