Question

$\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}$ is equal at
​
(a) naⁿ
(b) naⁿ⁻¹
(c) na
(d) 1

Answer 1

(b) naⁿ⁻¹

$$\begin{gathered} \underset{x\to a}{\lim }\frac{x^n-a^n}{x-a} \\ =\underset{x\to a^{+}}{\lim }\frac{x^n-a^n}{x-a}\left[\because f\left(x\right)\mathrm{exists},\underset{x\to a}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)\right] \\ =\underset{h\to 0}{\lim }\frac{{\left(a+h\right)}^n-a^n}{a+h-a} \\ =\underset{h\to 0}{\lim }{a}^n\frac{\left[{\left(1+{\displaystyle \frac{h}{a}}\right)}^n-1\right]}{h} \\ =a^n\underset{h\to 0}{\lim }\left[1+n·\frac{h}{a}+\frac{n\left(n-1\right)}{2!}\frac{h^2}{a^2}...+...-1\right] \\ =a^n\underset{h\to 0}{\lim }\left[\frac{n}{a}+\frac{h\left(h-1\right)}{2!}\frac{h}{a^2}+...\right] \\ =a^n\frac{n}{a} \\ =n{a}^{n-1} \end{gathered}$$