# show that $(\frac{\sqrt{1+x+x^2}-1}{\sqrt{1+x}-\sqrt{1-x}})$ tends to a limit as $x \rightarrow 0$, and evaluate the limit.

Show that $$(\frac{\sqrt{1+x+x^2}-1}{\sqrt{1+x}-\sqrt{1-x}})$$ tends to a limit as $$x \rightarrow 0$$, and evaluate the limit.

Taking the L'Hopital's rule, I get $$\frac{\frac{2x+1}{\sqrt{1+x+x^2}}}{(\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1-x}})}$$

So as $$x \rightarrow 0$$, the equation tends $$1/2$$.

Is this correct?

The problem is that $$\sqrt{1+x+x^2}$$ is not zero when x tends to zero, it means that you have no an undetermined term of the form $$0/0$$ to justify the application of the L'Hopital theorem. You should replace at first and check that the expression is suitable to apply the L'Hopital rule if not you will get a wrong calculation of the limit.
• In this case, your answer $1/2$ is right and your procedure is right also. Mar 24 '20 at 17:48