题目

求下列极限.-|||-lim _(xarrow -1)(dfrac (1)(x+1)-dfrac (4)({x)^3+1})

$\begin{aligned}&\text{求下列极限.}\\ &\lim_{x\to-1}\left(\frac{1}{x+1}-\frac{4}{x^{3}+1}\right) \\&\end{aligned}$

题目解答

答案

$\begin{aligned}&\\ &\\ & \lim_{x\to -1}\left(\frac{1}{x + 1} - \frac{4}{x^3 + 1}\right)\\ & =\lim_{x\to -1}\left(\frac{1}{x + 1} - \frac{4}{(x + 1)(x^2 - x + 1)}\right)\\ & =\lim_{x\to -1}\left(\frac{x^2 - x + 1 - 4}{(x + 1)(x^2 - x + 1)}\right)\\ & =\lim_{x\to -1}\left(\frac{x^2 - x - 3}{(x + 1)(x^2 - x + 1)}\right)\\ & =\lim_{x\to -1}\left(\frac{(x + 1)(x - 2)}{(x + 1)(x^2 - x + 1)}\right)\\ & =\lim_{x\to -1}\left(\frac{x - 2}{x^2 - x + 1}\right)\\ & =\frac{-1 - 2}{(-1)^2 - (-1) + 1}\\ & =\frac{-3}{3}\\ & =-1 \\ &\\ & \\&\end{aligned}$