Найти предел:
\(
\lim_{x\to0} \large \frac{a^x-l^x}{\tan{x}}.
\)

Решение

\(\displaylines{
\lim_{x\to0} \frac{a^x-l^x}{\tan{x}} =
= \lim_{x\to0} \frac{ a^x \ln a - l^x \ln l }{\frac{1}{\cos^2{x}} } =\\
= \lim_{x\to0} \Bigr( (a^x \ln a - l^x \ln l )\cos^2{x} \Bigl)
= \ln a - \ln e = \ln \frac{a}{e}}
\)

Ответ: \( \large \frac{a}{e} \).