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My aim is to evaluate the following limit

$\displaystyle\lim _{ x\rightarrow 0 }{ \frac { \sin ^{ -1 }{ x } -\tan ^{ -1 }{ x } }{ { x }^{ 3 } } } \quad$

which evaluates to $\frac{1}{2}$ after using L'Hospital rule for rather $5$ steps, which of course is a tedious task. Could anyone suggest some shorter and elegant methods for evaluating this limit?

Thanks.

Das
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    Using $\arcsin'(x) = (1-x^2)^{-1/2}$ and $\arctan'(x) = (1+x^2)^{-1}$ and the generalized binomial theorem it's straight forward to expand these functions in a power-series. – Winther May 22 '18 at 11:47
  • Thanks, I was unaware of the expansions of inverse trigonometric functions. – Das May 22 '18 at 12:39
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    btw to derive the expression for the derivatives above you can consider $\arctan(\tan(x)) = x$, take the derivative and express the result in terms of $y = \tan(x)$. Similar for other trigonometric functions. – Winther May 22 '18 at 12:43
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    Write numerator as $-(x-\sin^{-1}x)+(x-\tan^{-1}x)$ and use https://math.stackexchange.com/questions/387333/are-all-limits-solvable-without-lh%C3%B4pital-rule-or-series-expansion – lab bhattacharjee May 25 '18 at 18:36

3 Answers3

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We have

$$ \sin(\arcsin(x)-\arctan(x))=\frac{x(1-\sqrt{1-x^2})}{\sqrt{1+x^2}} $$

and

$$ \lim_{x\to 0}\left(\frac{\arcsin(x)-\arctan(x)}{x^3}\right)\equiv \lim_{x\to 0}\left(\frac{\sin(\arcsin(x)-\arctan(x))}{\sin(x^3)}\right) $$

then

$$ \frac{\sin(\arcsin(x)-\arctan(x))}{\sin(x^3)} = \frac{x^3(1-\sqrt{1-x^2})}{x^2\sqrt{1+x^2}\sin(x^3)} = \left(\frac{x^3}{\sin(x^3)}\right)\frac{x^2}{x^2(1+\sqrt{1-x^2})\sqrt{1+x^2}} $$

then

$$ \lim_{x\to 0}\left(\frac{\sin(\arcsin(x)-\arctan(x))}{x^3}\right)\equiv\lim_{x\to 0}\left(\frac{x^3}{\sin(x^3)}\right)\frac{1}{(1+\sqrt{1-x^2})\sqrt{1+x^2}} = \frac{1}{2} $$

Cesareo
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Using Maclaurin series:

$$\arcsin x-\arctan x=x+\frac16x^3+\frac3{40}x^5+\ldots-\left(x-\frac13x^3+\frac15x^5-\ldots\right)=$$

$$=\frac12x^3+\mathcal O(x^5)$$

so

$$\lim_{x\to0}\frac{\arcsin x-\arctan x}{x^3}=\lim_{x\to0}\frac{\frac12x^3+\mathcal O(x^5)}{x^3}=\frac12$$

DonAntonio
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    The use of series expansion is equivalent to the use of l'Hopital rule. – Cesareo May 22 '18 at 13:06
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    @CesarEo No, it's not...but it is very similar, true. Yet the OP did not want to avoid derivatives and L'Hospital, but rather to have a faster, simpler method to evaluate that limit. – DonAntonio May 22 '18 at 13:08
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This is easy!

Lets start by noting that $$arcsin(x)=x+x^3/6+3x^5/40+...$$ Next $$arctan(x)=x-x^3/3+x^5/5-...$$ When this is substituted in you limit equation above we have for the numerator: $$x^3/2-x^5/8 ...$$ When this is divided by $x^3$ and we pass to the limit we get the simple result of 1/2.