Question

If f and g are continuous functions with f(0) = 2 and the following limit, find g(0). lim_(x->0)[2 f(x) - g(x)] = -3

Answer 1

We have by distributivity of limits over sums, and continuity of $f(x)$ and $g(x)$, that

$\displaystyle \lim_{x\to0} \bigg(2f(x) - g(x)\bigg) = 2 \lim_{x\to0} f(x) - \lim_{x\to0} g(x) = 2 f(0) - g(0)$

Given $f(0)=2$, it follows that

$2^2 - g(0) = -3 \implies g(0) = \boxed{7}$