Найти предел:

\(
\lim_{x\to0} \large \frac{x-\arctan x}{x^3}.
\)

Решение

\(\displaylines{
\lim_{x\to0} \frac{x-\arctan x}{x^3} =
\lim_{x\to0} \frac{1-\frac{1}{1+x^2}}{3x^2} =
\lim_{x\to0} \frac{\frac{x^2}{1+x^2}}{3x^2} = \\
= \lim_{x\to0} \frac{x^2}{3x^2(1+x^2)} =
\lim_{x\to0} \frac{1}{3(1+x^2)} = \frac{1}{3}
}\)

Ответ: \( \large \frac{1}{3} \).