Question-3

Evaluate the following limit:

\[ \lim_{x \rightarrow \infty} \sqrt{x^2 + x + 1} - \sqrt{x^2 + 1} \]

Hint

Normalize the numerator.

Answer

\(0.5\)

Solution

\[ \begin{aligned} \lim_{x \rightarrow \infty} \sqrt{x^2 + x + 1} - \sqrt{x^2 + 1} &= \lim_{x \rightarrow \infty} \cfrac{(x^2 + x + 1) - (x^2 + 1)}{\sqrt{x^2 + x + 1} + \sqrt{x^2 + 1}}\\\\ &= \lim_{x \rightarrow \infty} \cfrac{x}{\sqrt{x^2 + x + 1} + \sqrt{x^2 + 1}}\\\\ &= \lim_{x \rightarrow \infty} \cfrac{1}{\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}} + \sqrt{1 + \frac{1}{x^2}}}\\\\ &= \cfrac{1}{2} \end{aligned} \]