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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}+7\text{x}}{4\text{x}+\sin2\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\frac{\sin3\text{x}}{\text{x}}+\frac{7\text{x}}{\text{x}}\big)}{\big(\frac{4\text{x}}{\text{x}}+\frac{\sin2\text{x}}{\text{x}}\big)}$
$=\frac{\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{3\text{x}}\times3\Big)+7}{4+\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\Big)\times2}$
$=\frac{3+7}{4+2}$
$=\frac{10}{6}$
$=\frac53$

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