Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow 0} \frac{\sin m x}{\tan n x}$

 

Solution:

To Find: Limits

NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.

In this Case, indeterminate Form is $\frac{0}{0}$

Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\operatorname{tanx}}{x}=1$

So $\lim _{x \rightarrow 0} \frac{\sin m x}{\tan n x}=\lim _{x \rightarrow 0}\left(\frac{\sin m x}{m x}\right) \times \frac{n x}{\tan n x} \times \frac{m x}{n x}=\frac{m x}{n x}=\frac{m}{n}$

Therefore, $\lim _{x \rightarrow 0} \frac{\sin m x}{\tan n x}=\frac{m}{n}$

 

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