Question

1. find the limit.

lim(x→-∞) (√(x² + 1))/(x + 1)

Explanation:

Step1: Divide numerator and denominator by -x

Since \(x\to-\infty\), we divide \(\frac{\sqrt{x^{2}+1}}{x + 1}\) by \(-x\) (because \(x<0\) as \(x\to-\infty\), and \(\sqrt{x^{2}}=-x\) for \(x < 0\)). We get \(\lim_{x\to-\infty}\frac{\frac{\sqrt{x^{2}+1}}{-x}}{\frac{x + 1}{-x}}=\lim_{x\to-\infty}\frac{\sqrt{\frac{x^{2}+1}{x^{2}}}}{- 1-\frac{1}{x}}\).

Step2: Simplify the expression inside the square - root

\(\frac{x^{2}+1}{x^{2}} = 1+\frac{1}{x^{2}}\), so the limit becomes \(\lim_{x\to-\infty}\frac{\sqrt{1+\frac{1}{x^{2}}}}{-1-\frac{1}{x}}\).

Step3: Use the limit rules

We know that \(\lim_{x\to-\infty}\frac{1}{x}=0\) and \(\lim_{x\to-\infty}\frac{1}{x^{2}} = 0\). Substituting these values into the expression \(\frac{\sqrt{1+\frac{1}{x^{2}}}}{-1-\frac{1}{x}}\), we have \(\frac{\sqrt{1 + 0}}{-1-0}\).

Answer:

\(-1\)