Limit using definition of $e$ I'm trying to calculate the following limit:
$$
\lim_{x\to0}\bigl(-4x + \sqrt{16x^2 + 1}\,\bigr)^{b/x}
$$
My instinct is as follows: use the binomial theorem on the square root, cancel higher order terms (i.e treat the square root as $1$), then appeal to the limit definition of $e$ to get the answer $\exp(-4b)$. However, this does not feel terribly rigorous and I'm unsure if it's right.
Does anyone have any advice?

You can use little-$o$ notation. You're taking the limit of $$(1-4x+8x^2+o(x^2))^{b/x}=\exp \frac{b}{x}(-4x+o(x))=\exp (-4b+o(1))=\exp(-4b)+o(1).$$
