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Factorization Method Form to Remove Indeterminate Form

Prove it L =...

Question

Prove it
$L = lim x \to 1 \frac{(x^{2} + 1) sin (x - 1)}{x^{2} - 3 x + 2}$

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Solution

${lim}_{x \to 1} \frac{(x^{2} + 1) sin (x - 1)}{x^{2} - 3 x + 2}$

$= {lim}_{x \to 1} \frac{(x^{2} + 1) sin (x - 1)}{x^{2} - 2 x - x + 2}$

$= {lim}_{x \to 1} \frac{(x^{2} + 1) sin (x - 1)}{x (x - 2) - (x - 2)}$

$= {lim}_{x \to 1} \frac{(x^{2} + 1) sin (x - 1)}{(x - 2) (x - 1)}$

$= {lim}_{x \to 1} \frac{x^{2} + 1}{x - 2} \times \frac{sin (x - 1)}{(x - 1)}$

$= \frac{1 + 1}{1 - 2} \times 1$

$= - \frac{2}{1} = - 2$

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