How to calculate limit without L'Hopital

How can I calculate limit without using L’Hopital rule?

$\displaystyle\lim_{x\to 0 }\frac{e^{\arctan(x)}-e^{\arcsin(x)}}{1-\cos^3(x)}$.

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Let $\arctan x = t$ so that $x = \tan t$ so that $\sin t = x/\sqrt{1 + x^{2}}$ or $\arctan x = t = \arcsin (x/\sqrt{1 + x^{2}})$.

Now we have

$\displaystyle\begin{aligned}\arctan x – \arcsin x &= \arcsin (x/\sqrt{1 + x^{2}}) – \arcsin x\\
&= \arcsin\left(x\sqrt{\frac{1 – x^{2}}{1 + x^{2}}} – \frac{x}{\sqrt{1 + x^{2}}}\right)\\
&= \arcsin\left\{\frac{x}{\sqrt{1 + x^{2}}}\left(\sqrt{1 – x^{2}} – 1\right)\right\}\\
&= \arcsin\left\{\frac{-x^{3}}{\sqrt{1 + x^{2}}\left(\sqrt{1 – x^{2}} + 1\right)}\right\}\\
&= \arcsin y = f(x) \text{ (say)}\end{aligned}$

Next we have

$\displaystyle\begin{aligned}\lim_{x \to 0}\frac{e^{\arctan x} – e^{\arcsin x}}{1 – \cos^{3}x} &= \lim_{x \to 0}e^{\arcsin x}\cdot\frac{e^{f(x)} – 1}{1 – \cos^{3} x}\\
&= \lim_{x \to 0}1\cdot\frac{e^{f(x)} – 1}{f(x)}\cdot\frac{f(x)}{1 – \cos^{3}x}\\
&= \lim_{x \to 0}1\cdot 1\cdot\frac{f(x)}{(1 – \cos x)(1 + \cos x + \cos^{2}x)}\\
&= \frac{1}{3}\lim_{x \to 0}\frac{f(x)}{1 – \cos x}\\
&= \frac{1}{3}\lim_{x \to 0}\frac{\arcsin y}{y}\cdot \frac{y}{1 – \cos x}\\
&= \frac{1}{3}\lim_{x \to 0}\frac{y}{2\sin^{2}(x/2)}\\
&= \frac{1}{6}\lim_{x \to 0}\frac{y}{(x/2)^{2}}\cdot \frac{(x/2)^{2}}{\sin^{2}(x/2)}\\
&= \frac{1}{6}\lim_{x \to 0}\frac{y}{(x/2)^{2}}\cdot 1\\
&= \frac{2}{3}\lim_{x \to 0}\frac{y}{x^{2}}\\
&= \frac{2}{3}\lim_{x \to 0}\frac{-x}{\sqrt{1 + x^{2}}\left(\sqrt{1 – x^{2}} + 1\right)}\\
&= \frac{2}{3}\cdot 0 = 0\end{aligned}$

No Taylor or L’Hospital is required. We just need algebraic and trigonometric manipulation combined with the use of standard limits.

Hint: remember that (from taylor series) $$e^x \sim 1 + x$$ if $x \to 0$, and $$\cos x \sim 1 – \frac{x^2}{2}$$ if $x \to 0$

Substitute into your fraction and you should get to the result easily.

The only thing that you need are the taylor expansion of arctan and arcsin

$$\arcsin x \sim x + \frac{x^3}{6}$$
$$\arctan x \sim x – \frac{x^3}{3}$$

Always with $x \to 0$

Just substitute and you’re good to go

(note that this are taylor expansion limited at the third order)

(by the way, be careful with this method unless you understand what $o(x)$ means (I omitted it in my answer) and how to use it.. for example it’s true that $\sin x \sim x$, but you can’t write $(\sin x – x) \sim (x – x) \sim 0$, it just has no meaning. In such cases you should use the sin approximation of superior order. $\sin x \sim x – \frac{x^3}{3} \Rightarrow (\sin x – x) \sim (x – \frac{x^3}{3} – x) \sim -\frac{x^3}{3}$ )

arctan x ~ x-1/3(x^3)+…
arcsin x ~ x+1/6(x^3)+…
e^ arctan x -e^ arcsin x ~1+ x-1/3(x^3) -(1+x+1/6(x^3) )~ -3 x^3 /6
1-cos^3 x ~ (1-cos x) (1+cos x +cos^2x )~ x^2 /2 (1+cos x+cos^2x)
put them in limit
you will have lim =0