Evaluate the following limits: _ x 3x^ -1 +4x^ -2 5x^ -1 +6x^ -2 - Vidyadip

Question

Evaluate the following limits:
$\lim\limits_{\text{x}\rightarrow\infty}\frac{3\text{x}^{-1}+4\text{x}^{-2}}{5\text{x}^{-1}+6\text{x}^{-2}}$

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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\frac{3\text{x}^{-1}+4\text{x}^{-2}}{5\text{x}^{-1}+6\text{x}^{-2}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{\frac{3}{\text{x}}+\frac{4}{\text{x}^2}}{\frac{5}{\text{x}}+\frac{6}{\text{x}^2}}$ $\Big[\frac{0}{0}\text{ from}\Big]$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{\frac{1}{\text{x}}\big(3+\frac{4}{\text{x}}\big)}{\frac{1}{\text{x}}\big(5+\frac{6}{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{(3+0)}{(5+0)}=\frac35$

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