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Problem 25 Medium Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Answer

This limit has the form $\frac{0}{0}$
\[
\begin{aligned}
\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1-4 x}}{x} & \underline{\Perp} \underset{x \rightarrow 0}{\lim } \frac{\frac{1}{2}(1+2 x)^{-1 / 2} \cdot 2-\frac{1}{2}(1-4 x)^{-1 / 2}(-4)}{1} \\
&=\lim _{x \rightarrow 0}\left(\frac{1}{\sqrt{1+2 x}}+\frac{2}{\sqrt{1-4 x}}\right)=\frac{1}{\sqrt{1}}+\frac{2}{\sqrt{1}}=3
\end{aligned}
\]

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Video Transcript

find the limit. Used a locked US rule where a proper it. If there's a moral dimension, my third consider using it. Okay, here, this is a fraction. So first thing to find the limit off this function is plucking zero to this function. And if he plucking zero, hear that hiss square road off ramp Last zero, which is one and minus squared off one minus zero, which is also one he wanted by zero. So this is zero divided by zero, which is a determined form so we can use the lob social. Okay, so there's limit is as X goes to zero, that purity of off the top, which is that you're a tip off the first of function, minus that they were here for the second one. Okay, the first function because that we can use the power room first. So that hiss one plus two acts to the power of a half if he used the power rule became put the power in front and plus two X through the power off original power minus one, which is minus a half and then used the shrine room to do that. The earth here, Father inside function, which is one class two acts, and this is a linear function. So the curative off it is too. So the times to hear and then minus used the same method to find a cure. It half off the second second term, that is a half times one minus here, X to the power minus half. And then times the curative off the inside function, which is minus four. So plus four. Okay. And then through the curative off the denominator, Which is what? Okay, let's do the computation off the top. So Lim as X goes to zero. And here this is one. Right. So we right here, this here the denominators want the talk. It's one plus to X to the power off. Negative half plus two times. Well, I'm minus four x to the power off a half and now we can plug in zero here. So this is one plus zero to a powerful like till a half. And this one and then plus two times. Well, I'm minus full time zero and to the power of ninety or half, which is also one. So that his two and then divided by one. So the answer is three