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I am unable to solve following limit: $$\lim_{x \rightarrow 0_+}\frac{\sin \sqrt{x}}{x^2}\left(\sqrt{x+2x^2}-\sqrt{2\sqrt{1+x}-2}\right)$$ I keep getting $ \frac{0}{0}$. I admit I haven't tried to use l'Hospital rule multiple times as the square root is not so nice to derive more than one time. Is it possible to solve this limit without using l'Hospital rule/Taylor series (which I haven't learned yet)? Thank you

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Try multiplying the top and bottom by the conjugate. –  gekkostate Jan 20 at 16:14
    
@cryogenic : I would start by replacing $x$ with $t^2$ to get rid of the square root. Use $\sin(t)/t \to 1$ as $t \to 0$ (which does not require either of the two things you mentioned). No more $\sin$. –  Stefan Smith Jan 20 at 16:18
    
ok so if i replace x^2 by t, I also have to add another square roots into the existing square roots? Or what you are trying to suggest is that I "divide" x^2 to have $\frac{\sin{\sqrt{x}}}{\sqrt{x}}\codt \frac{1}{x\sqrt{x}}$? –  D.N Jan 20 at 16:19

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Note that $\sqrt{2\sqrt{1+x}-2}\sim\sqrt{x+2x^2}\sim\sqrt x$, while \begin{align} (x+2x^2)-(2\sqrt{1+x}-2) &=2x^2+x+2-2\sqrt{1+x}\\ &=\frac{(2x^2+x+2)^2-4(1+x)}{2x^2+x+2+2\sqrt{1+x}}\\ &\sim\frac 94x^2 \end{align} thus \begin{align} \frac{\sin \sqrt{x}}{x^2}(\sqrt{x+2x^2}-\sqrt{2\sqrt{1+x}-2}) &=\frac{\sin \sqrt{x}}{x^2}\frac{(x+2x^2)-(2\sqrt{1+x}-2)}{\sqrt{x+2x^2}+\sqrt{2\sqrt{1+x}-2}}\\ &\sim \frac{\sqrt x}{x^2}\frac{\frac 94x^2}{2\sqrt x}\\ &=\frac 98 \end{align}

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Thank you for answering, but this solution uses Taylor series right? –  D.N Jan 20 at 16:31
    
No, it's elementary, only asympthotic notation is required such as $\sin(x)\sim x$ and $\sqrt{1+x}\sim \frac 12x$. –  Fabio Lucchini Jan 20 at 16:33
    
I tried to google it, and it looks interesting, but is elementary to the extent that such thing would be covered in intro class of analysis? –  D.N Jan 20 at 16:39
    
Yes, it's simply a notation; for example $\sin(x)\sim x$ is another way to write $\lim_{x\to 0}\frac{\sin(x)}x=1$, while $\sqrt{1+x}-1\sim \frac 12x$ follows from $\frac{\sqrt{1+x}-1}x=\frac{1+x-1}{x(\sqrt{1+x}+1)}\to\frac 12$. –  Fabio Lucchini Jan 20 at 16:39
    
alright, I am going to go through your solution(it takes a lot of time for me) and I will comment later if I need some clarification. –  D.N Jan 20 at 16:46

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