Showing that $e^{\frac{x}{a}}\simeq \sqrt{\frac{a+x}{a-x}}$

Question

I'm trying to solve an exercise whose text reads thus:

Show that for $|x|\ll a$, to within $(\frac{x}{a})^2$, we have the approximate equality $$e^{\frac{x}{a}}\simeq \sqrt{\frac{a+x}{a-x}}$$

What I think the text means is that I should prove that $e^{\frac{x}{a}}- \sqrt{\frac{a+x}{a-x}}<(\frac{x}{a})^2 $ for sufficiently small values of $|x|$. Is it right? I don't really undestand what "to within" is supposed to mean (not a native english-speaker). In case my interpreation is correct, is my solution fine as well?

By using Taylor expansion, we derive the following equalities $$e^{\frac{x}{a}}=1+\frac{x}{a}+\frac{g''(\xi_1)}{2}x^2$$ $$ f(x)=\sqrt{\frac{a+x}{a-x}}=1+\frac{x}{a}+\frac{f''(\xi_2)}{2}x^2 $$ with $|\xi_{1,2}|<|x|$ Subtracting the two quantities we get $$ e^{\frac{x}{a}}- \sqrt{\frac{a+x}{a-x}}=\frac{g''(\xi_1)-f''(\xi_2)}{2}x^2 $$. Thus we have to prov that $\frac{g''(\xi_1)-f''(\xi_2)}{2}<\frac{1}{a^2} $. Considering that $$g''(\xi_2)=\frac{a(2\xi_2 +a)}{(a+\xi_2)^{\frac{3}{2}}(a-\xi_2)^{\frac{5}{2}}}>0$$, since $|\xi_2| \ll a$, it is sufficient to prove that $$\frac{g''(\xi_1)}{2}= \frac{e^{\frac{\xi_1}{a}}}{2a^2}\leq \frac{1}{a^2} $$, which holds for $\xi_1<a\log{2}$, hence the thesis.

Answer 1

$$\log\left(\frac{1+y}{1-y}\right)=\int_0^y\frac{2\,dt}{1-t^2}=\int_0^y\left(2+\frac{2t^2}{1-t^2}\right)\,dt=\int_0^y(2+O(t^2))\,dt=2y+O(y^3)$$

Let $y=x/a\implies$

$$ \log\frac{1+x/a}{1-x/a}=\frac{2x}{a}+O((x/a)^3)\implies\frac{1}{2}\log\frac{a+x}{a-x}=\frac{x}{a}+O((x/a)^3)$$

Exponentiate both sides to see the desired result.

Answer 2

It is enough to prove that in a neighbourhood of the origin we have $e^{2x}\approx\frac{1+x}{1-x}$, or $$ x\approx \frac{1}{2}\log\left(\frac{1+x}{1-x}\right) = \text{arctanh}(x) \tag{1}$$ that simply follows from $$ \text{arctanh}(x)=\sum_{n\geq 0}\frac{x^{2n+1}}{2n+1}=x+O(x^3).\tag{2} $$