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How do I as precisely as possible prove that the following limit goes to infinity?

$$\lim_{x\to 0}\frac {\sqrt{x^2+x+1}-1}{\sin(2x)}=\infty $$

It seems difficult. I have started the proof by selecting an $M>0$ and attempting to show that the function is $M$ is always greater than the function. My problem seems to be algebraically manipulating the function so that I can extract $|x|$.

1 Answers 1

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HINT:

The limit is not $\infty$. Note that

$$\begin{align} \frac{\sqrt{x^2+x+1}-1}{\sin(2x)}&=\left(\frac{\sqrt{x^2+x+1}-1}{\sin(2x)}\right)\left(\frac{\sqrt{x^2+x+1}+1}{\sqrt{x^2+x+1}+1}\right)\\\\ &=\frac{x(x+1)}{\sin(2x)}\,\frac{1}{\sqrt{x^2+x+1}+1} \end{align}$$

Now, use $\lim_{\theta \to 0}\frac{\sin(\theta)}{\theta}=1$.

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    Maybe Dole didn't know this: $\sin(2x) = 2\sin(x)\cos(x)$. – 2017-02-15
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    @Hopeless That identity isn't required. Just let $\theta =2x$. – 2017-02-15
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    Didn't see that way. Thanks! – 2017-02-15
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    @Hopeless You're welcome. My pleasure. – 2017-02-15
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    The real issue is that I made a sign error somewhere that made it seem like the limit was infinity on my graphic calculator. Thank you very much... – 2017-02-15
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    You're quite welcome. I'm pleased to hear that you have it now! – 2017-02-15