1. Tardigrade
  2. Question
  3. Mathematics
  4. Evaluate: displaystyle limx → 0 (√1 + x +x2 -1/x)

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Evaluate : $\displaystyle\lim_{x \to 0} \frac{\sqrt{1 + x +x^2} -1}{x}$

Limits and Derivatives

Solution:

$\displaystyle\lim_{x\to0} \frac{\sqrt{1+x +x^{2}} -1}{x} $
$= \displaystyle\lim _{x\to 0} \frac{\sqrt{1+x+x^{2}} -1}{x}. \frac{\sqrt{1+x+x^{2}} +1}{\sqrt{1+x+x^{2}} +1} $
$=\displaystyle\lim _{x\to 0} \frac{1+x+x^{2} -1}{x\left(\sqrt{1+x+x^{2}} +1\right)} $
$= \displaystyle\lim_{x\to0} \frac{x\left(1 +x\right)}{x\left(\sqrt{1+x+x^{2} } +1\right)}$
$ = \displaystyle\lim _{x\to 0} \frac{1+x}{\sqrt{1+x+x^{2}} +1} =\frac{1}{2}$