lim x→−1 x^2+1/(x^2+x)(x^3+1) bằng:

Question

\({\mathop{\lim }\limits_{x\to -1}} \frac{x^{2} +1}{\left(x^{2} +x\right)\left(x^{3} +1\right)}\) bằng: 

\(A.+\infty . \)

\(B.2. \)

\(C.-2. \)

\(D.-\infty .\)

Answer 1

Chọn D

Ta có\({\mathop{\lim }\limits_{x\to -1}} \frac{x^{2} +1}{\left(x^{2} +x\right)\left(x^{3} +1\right)} ={\mathop{\lim }\limits_{x\to -1}} \frac{x^{2} +1}{x\left(x+1\right)\left(x+1\right)\left(x^{2} -x+1\right)} \) ,

\(={\mathop{\lim }\limits_{x\to -1}} \frac{x^{2} +1}{x\left(x+1\right)^{2} \left(x^{2} -x+1\right)} =-\infty\)

vì\({\mathop{\lim }\limits_{x\to -1}} \left(x^{2} +1\right)=2, {\mathop{\lim }\limits_{x\to -1}} \left(x\left(x+1\right)^{2} \left(x^{2} -x+1\right)\right)=0\) 

và \(x\left(x+1\right)^{2} \left(x^{2} -x+1\right)<0,\; \forall x\in \left(-2\, ;\, 0\right)\backslash \left\{-1\right\}\)