How to obtain: $\lim_{x \to -\infty}\big(\sqrt{x^2+3x+2}+x-1\big)$?

Question

Please help me to do my homework Take look to the below image I would like a solution step by step of $$ \lim_{x \to -\infty}\left(\sqrt{x^2+3x+2}+x-1\right). $$ enter image description here

Answer 1

Hint. One may write, as $x \to -\infty$, $$ \begin{align} \sqrt{x^2+3x+2}+x-1&=\frac{(x^2+3x+2)-(x-1)^2}{\sqrt{x^2+3x+2}-(x-1)} \\\\&=\frac{5+1/x}{-\sqrt{1+3/x+2/x^2}-1+1/x} \end{align} $$ where we have used $\sqrt{x^2}=|x|=-x$ since $x<0$.

Answer 2

Similarly completing the square of $x^2+3x+2=(x+\frac{3}{2})^2-\frac{1}{4}$ as we go to $-\infty$ we get that the square root of this is equivalent to $-x-\frac{3}{2}$ because the $-\frac{1}{4}$ isn't going to matter as we go to $-\infty$ now our limit is $\lim_{x\to-\infty} -x-\frac{3}{2}+x-1$ which is then just $-\frac{5}{2}$