Evaluate the following limit: _ x -1 x^3+1 x+1 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^3+1}{{\text{x}+1}}$

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Answer

$\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^3+1}{{\text{x}+1}}$
$=\lim\limits_{\text{x}\rightarrow-1}\frac{(\text{x}+1)\big(\text{x}^2-\text{x}+1\big)}{(\text{x}+1)}$ $\Big[\text{a}^3+\text{b}^3=(\text{a}+\text{b})\big(\text{a}^2+\text{b}^2-\text{ab}\big)\Big]$
$=\lim\limits_{\text{x}\rightarrow-1}{(\text{x}^2-\text{x}-1)}$
$=(-1)^2-(-1)+1$
$=1+1+1$
$=3$

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