COMEDK · Maths · 25. Continuity and Differentiability
If \(f(x)=\left\{\begin{array}{ll}\dfrac{1-x^m}{1-x} & \text { if } x \neq 1 \\ 2 m-1 & \text { if } x=1\end{array}\right.\) and the function is discontinuous at \(x=1\), then
- A \(m \neq \dfrac{1}{2}\)
- B \(m \neq 1\)
- C \(m=1\)
- D \(m=\dfrac{1}{2}\)
Answer & Solution
Correct Answer
(B) \(m \neq 1\)
Step-by-step Solution
Detailed explanation
A function \(f(x)\) is continuous at \(x=a\) if \(\lim_{x \to a} f(x) = f(a)\). Given \(f(x) = \dfrac{1-x^m}{1-x}\) for \(x \neq 1\), we calculate the limit as \(x \to 1\): \(\lim_{x \to 1} f(x) = \lim_{x \to 1} \dfrac{1-x^m}{1-x}\). Using L'Hopital's rule or the standard limit…
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