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}^3-(-1)^3}{\text{x}-(-1)}$ [Dividing numerator and denominator by x - 1]
Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$
Here, n = 3, a = -1
$\Rightarrow\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^3-(-1)^3}{\text{x}-(-1)}=\text{na}^{\text{n}-1}$
$=3(-1)^{3-1}$
$=3(-1)^2$
$=3$

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