# Solve $\lim_{x \to 0} \frac{\sqrt{1+2x} - \sqrt{1-4x}}{x}$ without L'Hospital's Rule.

I need to solve $$\lim_{x \to 0} \frac{\sqrt{1+2x} - \sqrt{1-4x}}{x}$$ without using L'Hospital's Rule. Using that rule I found the equation becomes $$\lim_{x \to 0}(\frac{1}{\sqrt{1+2x}} - \frac{2}{\sqrt{1-4x}}) = \frac{-1}{\sqrt{1}}$$.

However, I'm not sure how to solve this without L'Hospital's Rule. The only tool I know of is multiply both numerator and denominator by a conjugate, but multiplying by $$\frac{\sqrt{1+2x} + \sqrt{1-4x}}{\sqrt{1+2x} + \sqrt{1-4x}}$$ doesn't seem to get me the same answer.

$$\lim_{x \to 0}\frac{\sqrt{1+2x} - \sqrt{1-4x}}{x} \cdot \frac{\sqrt{1+2x} + \sqrt{1-4x}}{\sqrt{1+2x} + \sqrt{1-4x}}$$

$$= \lim_{x \to 0} \frac{1+2x-(1-4x)}{x(\sqrt{1+2x} + \sqrt{1-4x})}$$

$$= \lim_{x \to 0} \frac{6x}{x(\sqrt{1+2x} + \sqrt{1-4x})}$$

$$= \lim_{x \to 0} \frac{6}{\sqrt{1+2x} + \sqrt{1-4x}}$$

$$=\frac{6}{\sqrt{1} + \sqrt{1}}$$

But $$\frac{6}{\sqrt{1} + \sqrt{1}} \neq \frac{-1}{\sqrt{1}}$$.

• Your answer is actually correct : $\frac 6{\sqrt 1 + \sqrt 1} = 3$ is correct. This means there is something wrong with how you got $\frac {-1}{\sqrt 1} = -1$, so you have made some error while doing L'Hopital it seems! Rationalizing is certainly the first thought coming to mind if you aren't doing L'Hopital in this case, so "the only tool you know" is good enough here There are not too many other rules around, really, at the calc 1 level that you need to know. Apr 5 '19 at 16:34
• @астонвіллаолофмэллбэрг Oh! okay, that changes things. I'll take a look at how I got $\frac{-1}{\sqrt{1}}$ Apr 5 '19 at 16:40
• Yes. Post that on the question if you cannot figure out your mistake, we'll do it together! Apr 5 '19 at 16:41
• @астонвіллаолофмэллбэрг I got it. My mistake was that I thought $\frac{d}{dx}\sqrt{1-4x} = \frac{2}{\sqrt{1-4x}}$ but it was really $\frac{-2}{\sqrt{1-4x}}$. Apr 5 '19 at 16:46
• All right good to know that it's been solved Apr 5 '19 at 18:32

$$\frac{\sqrt{1+2x}-\sqrt{1-4x}}{x} =\frac{1+x + o(x)-1 + 2x + o(x)}{x} = 3 + o(1)$$

So the limit is $$3$$.

• I am only a calc 1 student, and I'm not really advanced enough to understand what you're doing here. Could you please explain it a little more specifically? I don't know what the $o(x)$ and $o(1)$ notation represent. Apr 5 '19 at 16:35
• @LuminousNutria it is very simple fact, you can check that in wikipedia 'Big o notion' and see the example there.: ) Apr 5 '19 at 16:42

There is a mistake, it must be $$\frac{1}{\sqrt{1+2x}}+\frac{2}{\sqrt{1-4x}}$$