Question

Solve $\underset{x\to -1}{lim}\frac{x^2-x-2}{\left(x^2+x\right)+\sin \left(x+1\right)}$

Answer 1

$\underset{x\to -1}{lim}\frac{x^2-x-2}{\left(x^2+x\right)+\sin \left(x+1\right)}$

it is of the form $\frac{0}{0}$

So, we will apply L Hospital rule until we will get determinate form

Differentiating numerator and denominator we will get,

${lim}_{x\to -1}\frac{2x-1}{2x+1+\cos (x+1)}$

Differentiating numerator and denominator again we will get,

${lim}_{x\to -1}\frac{2}{2-\sin (x+1)}=1$