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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives2 Marks
Question
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{{{(X + 1)}^5} - 1}}{x}$
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Answer

Here $\mathop {\lim }\limits_{x \to 0} \frac{{{{(X + 1)}^5} - 1}}{x}$
$= \mathop {\lim }\limits_{x \to 0} \frac{{{{(X + 1)}^5} - 1}}{{(x + 1) - 1}}$
Putting x + 1 = y, as $x \to 0,\;y \to 1$
$\therefore \;\mathop {\lim }\limits_{y \to 0} \frac{{{y^5} - 1}}{{y - 1}} = 5.{(1)^{5 - 1}}$
$= 5 \times 1 = 5\left[ {\because \;\mathop {\lim }\limits_{x \to a} \frac{{{x^n} - {a^n}}}{{x - a}} = n \cdot {a^{n - 1}}} \right]$

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