# Lim X → a X N − a N X − a is Equal at - Mathematics

MCQ

$\lim_{x \to a} \frac{x^n - a^n}{x - a}$  is equal at

• na

• nan−1

• na

•  1

#### Solution

nan−1

$\lim_{x \to a} \frac{x^n - a^n}{x - a}$
$= \lim_{x \to a^+} \frac{x^n - a^n}{x - a} \left[ \because f\left( x \right) exists, \lim_{x \to a} f\left( x \right) = \lim_{x \to a^+} f\left( x \right) \right]$
$= \lim_{h \to 0} \frac{\left( a + h \right)^n - a^n}{a + h - a}$
$= \lim_{h \to 0} a^n \frac{\left[ \left( 1 + \frac{h}{a} \right)^n - 1 \right]}{h}$
$= a^n \lim_{h \to 0} \left[ 1 + n \cdot \frac{h}{a} + \frac{n\left( n - 1 \right)}{2!}\frac{h^2}{a^2} . . . + . . . - 1 \right]$
$= a^n \lim_{h \to 0} \left[ \frac{n}{a} + \frac{h\left( h - 1 \right)}{2!} \frac{h}{a^2} + . . . \right]$
$= a^n \frac{n}{a}$
$= n a^{n - 1}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 11 | Page 78