Задача №2031
Условие
Найти предел \(\lim_{x\to{0}}\frac{x\left(e^x+1\right)-2\left(e^x-1\right)}{x^3}\).
Решение
\[
\lim_{x\to{0}}\frac{x\left(e^x+1\right)-2\left(e^x-1\right)}{x^3}
=\left[\frac{0}{0}\right]
=\lim_{x\to{0}}\frac{\left(x\left(e^x+1\right)-2\left(e^x-1\right)\right)'}{\left(x^3\right)'}=\\
=\lim_{x\to{0}}\frac{xe^x-e^x+1}{3x^2}
=\left[\frac{0}{0}\right]
=\lim_{x\to{0}}\frac{\left(xe^x-e^x+1\right)'}{\left(3x^2\right)'}
=\lim_{x\to{0}}\frac{xe^x}{6x}
=\lim_{x\to{0}}\frac{e^x}{6}
=\frac{1}{6}.
\]
Ответ:
\(\frac{1}{6}\)