Question

If $f(x)=\begin{cases}\frac{\log_{e}(1+\frac{x}{a})-\log_{e}(1-\frac{x}{b})}{x}& if~x\ne0\\k& if~x=0\end{cases}$ is continuous at $x=0$, then value of $k$ is:

• A. 0
• B. 1
• C. $\frac{1}{a}+\frac{1}{b}$
• D. $\frac{1}{a}-\frac{1}{b}$

Hint:

Using Taylor expansion for $\ln(1+u) \approx u - u^2/2 + \dots$ makes these limits very clear: the numerator is $(x/a) - (-x/b) = x(1/a + 1/b)$. Dividing by $x$ gives the result immediately.

Answer 1

The Correct Option is C