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.

  • $\begingroup$ sorry I forgot to add a -1 to the original problem $\endgroup$
    – spruce
    Mar 24 '20 at 17:34
  • 1
    $\begingroup$ In this case, your answer $1/2$ is right and your procedure is right also. $\endgroup$
    – Camilo160
    Mar 24 '20 at 17:48

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