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Substitution Method to Remove Indeterminate Form

limx→ 0sin x2...

Question

$lim x \to 0 \frac{sin (x^{2} - 5 x)}{x}$ .

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Solution

${lim}_{x \to 0} \frac{sin (x^{2} - 5 x)}{x} \times \frac{x^{2} - 5 x}{x^{2} - 5 x}$
${lim}_{x \to 0} \frac{sin (x^{2} - 5 x)}{x^{2} - 5 x} \times (x - 5) \to 1$
As $x \to 0, x^{2} - 5 x \to 0$
$\Rightarrow {lim}_{x^{2} - 5 x \to 0} \frac{sin (x^{2} - 5 x)}{{(x}^{2} - 5 x)} = 1$
$\Rightarrow {lim}_{x \to 0} 1 \times (x - 5)$ From $1$
$= 0 - 5 = - 5$

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