Evaluate the following limit: _ x 3 (x^2-9 ) (1 x+3 -1 x-3 ) - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow3}\big(\text{x}^2-9\big)\Big(\frac{1}{\text{x}+3}-\frac{1}{\text{x}-3}\Big)$

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Answer

$\lim\limits_{\text{x}\rightarrow3}\big(\text{x}^2-9\big)\Big(\frac{1}{\text{x}+3}-\frac{1}{\text{x}-3}\Big)$ $=\lim\limits_{\text{x}\rightarrow3}\big(\text{x}^2-9\big)\Big(\frac{\text{x}-3+\text{x}+3}{(\text{x}+3)(\text{x}-3)}\Big)$ $=\lim\limits_{\text{x}\rightarrow3}\big(\text{x}^2-9\big)\Big(\frac{2\text{x}}{(\text{x}+3)(\text{x}-3)}\Big)$ $=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}-3)(\text{x}+3)(2\text{x})}{(\text{x}+3)(\text{x}-3)}$ $=\lim\limits_{\text{x}\rightarrow3}2(\text{x})=2(3)=6$

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