Question

Examine the continuity of the functions
i) $f\left(x\right)=\{\begin{array}{ccc}\frac{x^n-1}{x-1},& \text{for}& x\ne 1\\ n^2,& \text{for}& x=1\end{array}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\frac{x^n-1}{x-1},x\ne 1\\ n^2,x=1\end{array}$

Checking continuity at $x=1$

$\Rightarrow \underset{x\to 1^{+}}{lim}\frac{x^n-1}{x-1}=\underset{h\to 0}{lim}\frac{{(1+n)}^n-1}{1+n-1}=\underset{h\to 0}{lim}\frac{{(1+n)}^{n-1}}{1}=n$ (L' Hospital Rule)

$\Rightarrow \underset{x\to 1^{-}}{lim}\frac{x^n-1}{x-1}=\underset{h\to 0}{lim}\frac{{(1-n)}^n-1}{1-n-1}=\underset{h\to 0}{lim}\frac{{(1-n)}^{n-1}(-1)}{-1}=n$ (L' Hospital Rule)

$\Rightarrow f\left(1\right)=n^2$

Here $\underset{x\to 1}{lim}f\left(x\right)\ne f\left(1\right)$

$\Rightarrow$ This is a removable discontinuity.