Evaluate _ x 0 ax bx , ;a, ;b 0 - Vidyadip

Question

Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{\sin bx}},\;a,\;b \ne 0$

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Answer

Here $\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{\sin bx}}$$= \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin ax}}{{ax}} \times ax \times \frac{1}{{\frac{{\sin bx}}{{bx}} \times bx}}} \right]$
$ = \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin ax}}{{ax}} \times \frac{1}{{\frac{{sinbx}}{{bx}}}} \times \frac{{ax}}{{bx}}} \right] = \frac{a}{b}$$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin ax}}{{ax}}\frac{1}{{\frac{{\sin bx}}{{bx}}}}} \right]$
$ = \frac{a}{b} \times 1 \times 1 = \frac{a}{b}$

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