Question

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

• A. $n{a}^n$
• B. $n{a}^{n-1}$
  
• C. na
• D. 1

Answer 1

The correct option is B

$n{a}^{n-1}$

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