Question

If \[ f(x) = \left[ \tan \left( \frac{\pi}{4} + x \right) \right]^{\frac{1}{x}} \quad \text{if} \quad x \neq 0 \] \[ f(x) = k \quad \text{if} \quad x = 0 \] is continuous at \( x = 0 \), then \( k = \)

• A. \( e \)
• B. \( \sqrt{e} \)
• C. \( e^2 \)
• D. \( e^4 \)

Hint:

When finding limits for expressions involving \( \tan \), use the small angle approximation for \( \tan \left( \frac{\pi}{4} + x \right) \).

Answer 1

The Correct Option is C

Step 1: Apply the continuity condition.
For the function to be continuous at \( x = 0 \), we need to have: \[ \lim_{x \to 0} f(x) = f(0) = k \] Now, we evaluate \( \lim_{x \to 0} \left[ \tan \left( \frac{\pi}{4} + x \right) \right]^{\frac{1}{x}} \).
Step 2: Simplify the expression.
Using the approximation \( \tan \left( \frac{\pi}{4} + x \right) \approx 1 + x \), we simplify the expression and find: \[ \lim_{x \to 0} \left( 1 + x \right)^{\frac{1}{x}} = e \]
Step 3: Conclusion.
Thus, for the function to be continuous at \( x = 0 \), we need \( k = e^2 \), corresponding to option (C).