Evaluate: _ x 3 x^ 3 +27 x^ 5 +243 - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow 3}\frac{\text{x}^{3}+27}{\text{x}^{5}+243}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow 3}\frac{\text{x}^{3}+27}{\text{x}^{5}+243}$
$=\lim\limits_{\text{x} \rightarrow 3}\frac{\frac{\text{x}^{3}+(3)^{3}}{\text{x}-3}}{\frac{\text{x}^{5}+(3)^{3}}{\text{x}-3}}$ [Dividing the Nr and Den. By x - 3]
$=\lim\limits_{\text{x} \rightarrow 3}\frac{\frac{\text{x}^{3}-(3)^{3}}{\text{x}+3}}{\frac{\text{x}^{5}-(3)^{5}}{\text{x}+3}}$
$=\frac{3(-3)^{3-1}}{5(-3)^{5-1}}$
$=\frac{3\times(-3)^{2}}{5\times(-3)^{4}}$
$=\frac{1}{5\times3}=\frac{1}{15}$
Hence, the required answer is $\frac{1}{15}.$

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