Practicing Success
Let $f(x)=\frac{\log _e(1+x)^{1+x}-x}{x^2}$, then find the value of f(0) so that the f(x) is continuous at x = 0. |
$\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$ $\frac{2}{3}$ |
$\frac{1}{2}$ |
Since f(x) is continuous at x = 0, we must have $f(0) =\lim\limits_{x \rightarrow 0} f(x)$ $=\lim\limits_{x \rightarrow 0} \frac{(1+x) \log _e(1+x)-x}{x^2}$ (form $\frac{0}{0}$) $=\lim\limits_{x \rightarrow 0} \frac{\log _e(1+x)+1-1}{2 x}$ (Using L' Hospital Rule) $=\frac{1}{2} \lim\limits_{x \rightarrow 0} \frac{\log _e(1+x)}{x}$ $=\frac{1}{2}$ |