I'm trying to evaluate $$\lim\limits_{b\to a}\frac{e^{-\frac{x}{a}}-e^{-\frac{x}{b}}}{a-b}$$
I know that the limit exists. The limit of the numerator and denominator are both zero when $b\to a$, so I tried to apply L'Hospital's Rule for $\frac00$ form, but the denominator is constant with respect to $x$ so it becomes zero and then the whole thing is undefined.
How does one go about tackling this limit?
Best Answer
This is nothing but
$$\frac{d}{da}e^{-\frac xa} = \frac x{a^2}e^{-\frac xa}$$
To see this, note that
$$\lim\limits_{b\to a}\frac{e^{-\frac{x}{a}}-e^{-\frac{x}{b}}}{a-b} = \lim\limits_{b\to a}\frac{e^{-\frac{x}{b}}-e^{-\frac{x}{a}}}{b-a}$$