# How to evaluate the following complex limit?

I have to evaluate the following limit:

$$\lim_{z\rightarrow \frac{\sqrt3i}{2}}\frac{z}{4z^2+3}$$

I know the limit doesn't exist. I tried proving it by letting: $$z = x +iy,\\y =\frac{\sqrt3i}{2}$$

And then evaluating the new limit for $$x\rightarrow0\_$$

and $$x\rightarrow0_+$$

and hence show they do not equal one another. However, this approach didn't work since I got to $$\lim_{x\rightarrow0\_}\frac{x+i\sqrt3/2}{4x(x+i\sqrt3)}$$

and got stuck.

Could I get some help guys?

Edit: sorry if the formatting isn't that good. I'm still new to this site.

• The expression has the form $c/0$ where $c\ne 0$. (And your development is wrong.) – Yves Daoust May 1 '17 at 19:46
• @YvesDaoust So, if it's simply c/0, it's enough to say the limit doesn't exist without any other proof? – Abdul Miah May 1 '17 at 19:52
• Yep, you are at a pole. – Yves Daoust May 1 '17 at 19:53
• First thing when you meet a limit is to evaluate the function ! – Yves Daoust May 1 '17 at 19:55
• Sorry, your development wasn't wrong, my bad. You are not stuck, when $x\to0$, your expression $\to\infty$. – Yves Daoust May 1 '17 at 19:58

We observe $\frac{\sqrt{3}i}{2}$ is a pole of first order of $\frac{z}{4z^2+3}$ and so the limit does not exist.