Evaluate the following limit: _ x 0 5x+4 3x 4 2x+7x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{5\text{x}+4\sin3\text{x}}{4\sin2\text{x}+7\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{5\text{x}+4\sin3\text{x}}{4\sin2\text{x}+7\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{5+\frac{4\sin3\text{x}}{\text{x}}}{\frac{4\sin2\text{x}}{\text{x}}+7}$
$=\frac{\lim\limits_{\text{x}\rightarrow0}5+4\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{3\text{x}}\times3}{4\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2+7}$
$=\frac{5+4\times1\times3}{4\times2+7}$
$=\frac{5+12}{8+7}$
$=\frac{17}{15}$

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