Evaluating $\lim_{x \to 0} \frac{a_1\exp(-b_1x^2)}{\sum_i a_i\exp(-b_ix^2)}$

Question

I am looking for the solution of the following limit:

$$\lim_{x \to 0} \frac{a_1\exp(-b_1x^2)}{\sum_i a_i\exp(-b_ix^2)}$$

Since $\lim_{y\to0}\exp(y) = 1$, is $\frac{a_1}{\sum_i a_i}$ really the correct solution for this problem?

Answer 1

Provided the limit exists for $f,g$, the latter being nonzero, we can say

$$\lim \limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim \limits_{x \to c} f(x)}{\lim \limits_{x \to c} g(x)}$$

Take

• $c = 0$
• $f(x) = a_1\exp(-b_1x^2)$
• $g(x) = \sum \limits_{i} a_i\exp(-b_ix^2) = \exp(-b_ix^2) \sum \limits_{i} a_i$.

Thus, clearly,

$$\begin{align} \lim \limits_{x \to c} f(x) &= a_1 \\ \lim \limits_{x \to c} g(x) &= \sum \limits_{i} a_i \end{align}$$

ultimately verifying your result, if the limit of the denominator can be guaranteed to be nonzero.