Question

Найти предел БЕЗ правила Лопиталя.

Answer 1

$\displaystyle\lim_{x \to 1}\dfrac{1-x}{\cos\dfrac{3\pi x}{2}}=\left\{\begin{array}{ccc}1-x=t\\ x=1-t\\ t\to 0\end{array}\right\}=\lim_{t \to 0}\frac{t}{\cos \left(\dfrac{3\pi}{2}(1-t)\right)}=\\ \\ \\ =\lim_{t \to 0}\frac{t}{\cos \left(\dfrac{3\pi}{2}-\dfrac{3\pi t}{2}\right)}=\lim_{t \to 0}\dfrac{t}{-\sin\dfrac{3\pi t}{2}}=\lim_{t \to 0}\dfrac{t}{-\dfrac{3\pi t}{2}}=-\dfrac{2}{3\pi}$