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Question
Discuss the continuity of the following functions at the indicated point(s):
Solution
Given:
Here,
\[\lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \frac{1 - x^n}{1 - x}\]
\[ \Rightarrow \lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \left[ \left( 1 - x \right)^{n - 1} + \ prescript{n}{}{C}_1 \left( 1 - x \right)^{n - 2} x + \ prescript{n}{}{C}_2 \left( 1 - x \right)^{n - 3} x^2 + . . . + \ prescript{n}{}{C}_{n - 1} \left( 1 - x \right)^0 x^{n - 1} \right]\]
Thus,
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