Question

Which of the following function is discontinuous at \(x = 0\) ?

• A. \(f(x) = (1+x)^{\frac{2}{x}},\) for \(x\neq 0\)
  \(= e^2,\) for \(x = 0\)
• B. \(f(x) = sinx-cosx,\) for \(x\neq 0\)
  \(= -1,\) for \(x = 0\)
• C. \(f(x) = \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1},\) for \(x\neq 0\)
  \(= -1,\) for \(x = 0\)
• D. \(f(x) = \frac{e^{5x}-e^{2x}}{sin3x},\) for \(x\neq 0\)
  \(= 1,\) for \(x = 0\)

Hint:

Compare left and right limits at \(x=0\) for each piece.

Answer 1

The Correct Option is C

Step 1: Look for a jump
The only function with $e^{1/x}$ behaves differently on each side of $0$.

Step 2: Compute one-sided limits
Right limit $=1$, left limit $=-1$. These differ, so (C) is discontinuous, while the other three have limits equal to their given values.

Final Answer:
Only (C) has unequal one-sided limits, option (C). \[ \boxed{\text{(C)}} \]