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How can I solve this limit without L'Hôpital's rule?

$$\begin{align}\lim\limits_{x \to 0} \frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^3}&=\lim\limits_{x \to 0} \frac{\tan x-\sin x}{x^3(\sqrt{1+\tan x}+\sqrt{1+\sin x})}\\&=\lim\limits_{x \to 0} \frac{\frac{\tan x-\sin x}{x^3}}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\end{align}$$

I can't proceed anymore from here.

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    $\begingroup$ $\frac{\tan(x)-\sin(x)}{x^3}=\frac{\sin(x)}{x}\frac{1-\cos(x)}{x^2}\frac{1}{\cos(x)}=\frac{\sin(x)}{x}\frac{\cos(0)-\cos(x)}{x^2}\frac{1}{\cos(x)}=\frac{\sin(x)}{x}\frac{2\sin(x/2)^2}{x^2}\frac{1}{\cos(x)}$ $\endgroup$
    – logarithm
    May 2, 2019 at 6:47
  • $\begingroup$ @logarithm Thanks! but I think this is easier: $\endgroup$
    – firia2000
    May 2, 2019 at 8:11
  • $\begingroup$ $\frac{1-cosx}{x^2}=\frac{(1-cosx)(1+cosx)}{x^2(1+cosx)}=\frac{sin^2x}{x^2(1+cosx)}=(\frac{sinx}{x})^2\frac{1}{1+cosx}$ $\endgroup$
    – firia2000
    May 2, 2019 at 8:11

4 Answers 4

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Hint:

$\frac{\tan x- \sin x}{x^3}=\frac{1}{ \cos x} \cdot \frac{\sin x}{x} \cdot \frac{1 - \cos x}{x^2}.$

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$$\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^3} =\frac{\tan x-\sin x}{x^3(\sqrt{1+\tan x}+\sqrt{1+\sin x})}.$$ From Maclaurin series $\sin x=x-x^3/6+O(x^5)$ and $\tan x=x+x^3/3+O(x^5)$. Therefore $$\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^3} \sim\frac{x^3/2}{x^3(\sqrt{1+\tan x}+\sqrt{1+\sin x})}$$ as $x\to0$, and so the limit is $1/4$.

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Hint: $$\tan x- \sin x=\frac{x^3}{3}-\frac{x^3}{3!}+......$$

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We can work asymptotically also: We denote the limit by $L$

As $x \to 0$

$$L=\frac{(\sqrt{1+\tan x}-1) -(\sqrt{1+\sin x}-1)}{x^3} \sim \frac{1}{2} \,\,\left[ \frac{\tan x (1-\cos x)}{x^3} \right] \sim \frac{1}{4}\,\,\frac{\tan x}{x} \sim \frac{1}{4}$$

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