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Q. $\displaystyle \lim_{x \to 1}$ $\frac{x^{3}+x^{2}-2}{sin\left(x-1\right)}=$

Limits and Derivatives

Solution:

$\displaystyle \lim_{x \to 1}$$\frac{\left(x-1\right)\left(x^{2}+2x+2\right)}{sin\left(x-1\right)}$
$=\displaystyle \lim_{x \to 1}$$\frac{x-1}{sin\left(x-1\right)} \displaystyle \lim_{x \to 1}$$\left(x^{2}+2x+2\right)$
$=1\left(1+2+2\right)=5$.