Question

a^x - b^x / x as x tends to 0

Answer 1

I suppose L'Hopital's Rule could be applied here, since both the numerator, and the denominator both to go 0 at x=0.
\[\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}\]In this case, the numerator will become \(a^xln(a) - b^xln(b)\), and the denominator will go to 1. So the limit becomes:\[\lim_{x \rightarrow 0} \frac{a^x-b^x}{x} = \lim_{x \rightarrow 0} \frac{a^xln(a)-b^xln(b)}{1}\]Which should be calculatable!