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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives1 Mark
Question
Evaluate: $\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}}$
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Answer

We have,
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}} \times \frac{{(x - 1)}}{{(x - 1)}}$
$= \mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{x - 1}} \div \mathop {\lim }\limits_{x \to 1} \frac{{{x^{10}} - 1}}{{x - 1}}$
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - {{(1)}^{15}}}}{{x - 1}} \div \mathop {\lim }\limits_{x \to 1} \frac{{{x^{10}} - {{(1)}^{10}}}}{{x - 1}}$
$= 15(1)^{14} \div 10(1)^9 [\because {\mathop {\lim }\limits_{x \to a} \frac{{{x^n} - {a^n}}}{{x - a}}} = na^{n - 1}]$
$= \frac { 15 } { 10 } = \frac { 3 } { 2 }$

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