Evaluate the following limits in Exercise: _ x 3 x^4-81 2x^2-5x-3 - Vidyadip

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^4-81}{2\text{x}^2-5\text{x-3}}$

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Answer

At x = 2, the value of the given rational function takes the form $\frac{0}{0}$.$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^4-81}{2\text{x}^2-5\text{x-3}}$$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}-3)(\text{x}+3)(\text{x}^2+9)}{(\text{x}-3)(2\text{x}+1)}$
$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}+3)(\text{x}^2+9)}{2\text{x}+1}$
$=\frac{(3+3)(3^2+9)}{2(3)+1}$
$=\frac{6\times18}{7}$
$=\frac{108}{7}$

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