Question

1. $lim_{x

ightarrow - 1}\frac{x^{3}+x^{2}-x - 1}{x^{3}+4x^{2}+5x + 2}$

Explanation:

Step1: Factor the numerator

$x^{3}+x^{2}-x - 1=x^{2}(x + 1)-(x + 1)=(x + 1)(x^{2}-1)=(x + 1)(x - 1)(x+1)$

Step2: Factor the denominator

$x^{3}+4x^{2}+5x + 2=x^{3}+x^{2}+3x^{2}+3x+2x + 2=x^{2}(x + 1)+3x(x + 1)+2(x + 1)=(x + 1)(x^{2}+3x + 2)=(x + 1)(x + 1)(x+2)$

Step3: Simplify the fraction

$\lim_{x
ightarrow - 1}\frac{(x + 1)(x - 1)(x + 1)}{(x + 1)(x + 1)(x + 2)}=\lim_{x
ightarrow - 1}\frac{x - 1}{x + 2}$

Step4: Substitute the limit value

$\frac{-1-1}{-1 + 2}=\frac{-2}{1}=-2$

Answer:

$-2$