_ x a x^n-a^n x-a is equal to: - Vidyadip

MCQ

$\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}$ is equal to:

✓
$n a^{ n -1}$
B
$1$
C
$na^n$
D
$na$

✓

Answer

Correct option: A.

$n a^{ n -1}$

$\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}$
$=\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}[\because f(x)$ exists, $\lim _{x \rightarrow a} f(x)$
$=\lim _{x \rightarrow a^{+}} f(x)]$
$=\lim _{h \rightarrow 0} \frac{(a+h)^n-a^n}{a+h-a}$
$=\lim _{h \rightarrow 0} a^n \frac{\left[\left(1+\frac{A}{a}\right)^n-1\right]}{h}$
$=a^{ n } \lim _{h \rightarrow 0}\left[1+ n \cdot \frac{h}{a}+\frac{n(n-1)}{2!} \frac{h^2}{a^2} \ldots+\ldots-1\right]$
$=a^{ n } \lim _{h \rightarrow 0}\left[\frac{n}{a}+\frac{h(h-1)}{2!} \frac{h}{a^2}+\ldots\right]$
$=a^{ n } \frac{n}{a}$
$=n a^{ n -1}$

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