Question

If \( f(x) = \begin{cases} \frac{1-x^m}{1-x}, & \text{for } x \neq 1 \\ 2m-1, & \text{for } x = 1 \end{cases} \) and the function is discontinuous at \( x = 1 \), then

• A. \( m=1 \)
• B. \( m\ne \frac{1}{2} \)
• C. \( m=\frac{1}{2} \)
• D. \( m\ne 1 \)

Hint:

To check discontinuity, first find the condition for continuity, then take its negation.

Answer 1

The Correct Option is D

Step 1: Check continuity condition.
For continuity at \(x=1\):
\[ \lim_{x\to 1} f(x) = f(1). \]

Step 2: Evaluate the limit.
\[ \lim_{x\to 1} \frac{1-x^m}{1-x}. \]
This is a standard limit:
\[ \lim_{x\to 1} \frac{1-x^m}{1-x} = m. \]

Step 3: Compare with given value.
Given:
\[ f(1) = 2m - 1. \]

Step 4: Apply continuity condition.
\[ m = 2m - 1. \]

Step 5: Solve equation.
\[ m = 1. \]

Step 6: Interpret discontinuity condition.
The function is discontinuous at \(x=1\), so the above condition must not hold.
Thus:
\[ m \ne 1. \]

Step 7: Final conclusion.
\[ \boxed{m \ne 1} \]