Đáp án:
\(\lim\limits_{x\to 0}\dfrac{2 - \sqrt{4 - x^2}}{1 - \sqrt{\arctan x^2 + 1}}= -\dfrac12\)
Giải thích các bước giải:
\(\begin{array}{l}
\quad \lim\limits_{x\to 0}\dfrac{2 - \sqrt{4 - x^2}}{1 - \sqrt{\arctan x^2 + 1}}\\
= \lim\limits_{x\to 0}\dfrac{x^2\left(1 + \sqrt{\arctan x^2 + 1}\right)}{-\arctan x^2\left(2 + \sqrt{4-x^2}\right)}\\
= \lim\limits_{x\to 0}\dfrac{x^2\left(1 + \sqrt{\arctan x^2 + 1}\right)}{-x^2\left(2 + \sqrt{4-x^2}\right)}\\
= -\lim\limits_{x\to 0}\dfrac{1 + \sqrt{\arctan x^2 + 1}}{2 + \sqrt{4-x^2}}\\
= - \dfrac{1 + \sqrt{\arctan 0^2 + 1}}{2 + \sqrt{4-0^2}}\\
= - \dfrac12
\end{array}\)
