1. ## It's limit.

Lim x->0 (e^sqrt(x) -2)/x
solve it.
Thank
I solved it but I've got 2 answers and I'm not sure. Sorry about my english. I'm not a native speaker .

2. ## Re: It's limit.

Originally Posted by Tiffany
Lim x->0 (e^sqrt(x) -2)/x solve it.
I solved it but I've got 2 answers and I'm not sure.
I hope the problem is actually ${\lim _{x \to {0^ + }}}\frac{{{e^{\sqrt x }} - 2}}{x}$, a one-sided limit.
Does that limit exist? Hint: if it does, it is not bounded.

3. ## Re: It's limit.

Sorry my bad. It is -1 not -2

4. ## Re: It's limit.

Originally Posted by Tiffany
Sorry my bad. It is -1 not -2
That does not change much. It still must be a one-sided limit.
The limit is unbounded.

5. ## Re: It's limit.

perhaps you were a bit over-eager to use l'hopital?

we can use l'hopital once, to get:

$\lim_{x \to 0^+} \frac{e^{\sqrt{x}} - 1}{x} = \lim_{x \to 0^+} \frac{e^{\sqrt{x}}}{2\sqrt{x}}$

but in the latter limit, the numerator goes to 1, while the denominator goes to 0, so the ratio is unbounded.

6. ## Re: It's limit.

If you allow complex values, then it is alright to use a two-sided limit but then if you are allowing such things, you might as well consider the limit on the complex plane. This might be a bit of a tangent but here we go. Let

$f(z) = \frac{e^{\sqrt{z}}-1}{z}$

for complex values $z$. We'll show that $f(z)$ is analytic. The condition with the Cauchy-Riemann equations is equivalent to saying that

$\frac{\partial f}{\partial \overline{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)f = 0$

for $z = x + i y$

If you calculate this, it is indeed equal to zero. So the function is analytic and we can take the limit as $z \to 0$. The limit happens to be real and unbounded in the positive direction.