Question

Discuss the continuity of the following functions at the indicated point(s): 

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{1 - x^n}{1 - x}, & x \neq 1 \\ n - 1 , & x = 1\end{array}n \in N \right.at x = 1\]

Answer 1

 Given:

\[f\left( x \right) = \binom{\frac{1 - x^n}{1 - x}, x \neq 1}{n - 1, x = 1}\]

Here, 

\[f\left( 1 \right) = n - 1\]

\[\lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \frac{1 - x^n}{1 - x}\]

\[ \Rightarrow \lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \left[ \left( 1 - x \right)^{n - 1} + \ prescript{n}{}{C}_1 \left( 1 - x \right)^{n - 2} x + \ prescript{n}{}{C}_2 \left( 1 - x \right)^{n - 3} x^2 + . . . + \ prescript{n}{}{C}_{n - 1} \left( 1 - x \right)^0 x^{n - 1} \right]\]

\[\Rightarrow \lim_{x \to 1} f\left( x \right) = 0 + 0 . . . + \left( 1 \right)^{n - 1} = 1 \neq f\left( 1 \right)\]

Thus, 

\[f\left( x \right) \text{is discontinuous at} x = 1\]