Evaluate: _ x 0 -2 3x+ 5x x - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}-2\sin3\text{x}+\sin5\text{x}}{\text{x}}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}-2\sin3\text{x}+\sin5\text{x}}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}-\frac{2\sin3\text{x}}{\text{x}}+\frac{\sin5\text{x}}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}-\lim\limits_{3\text{x} \rightarrow 0}2\big(\frac{\sin3\text{x}}{3\text{x}}\big)\times3+\lim\limits_{5\text{x} \rightarrow 0}\big(\frac{\sin5\text{x}}{5\text{x}}\big)\times5$
$=1-6+5=0$
Hence, the required answer is 0.

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