# Math Help - Complex Limit

1. ## Complex Limit

$\lim_{z\rightarrow -i}\frac{z^4-1}{z+ i}$

What do I do with a Complex limit when I get $\frac{0}{0}\mbox{?}$

2. Originally Posted by dwsmith
$\lim_{z\rightarrow -i}\frac{z^4-1}{z+ i}$

What do I do with a Complex limit when I get $\frac{0}{0}\mbox{?}$
We want

$\displaystyle \lim_{z \to -i}\frac{z^4-1}{z+i}=\lim_{z \to -i}\frac{(z^2-1)(z^2+1)}{z+1}=\lim_{z \to -i}\frac{(z^2-1)(z+i)(z-i)}{z+i}=...$

3. sorry as a side note I also saw the Hebrew chars, but I don't think latex needs the tag mathbb for i. I also just use the letter i, but then again maybe i am lazy.

4. I use the mathbb to bold the i.

5. Originally Posted by dwsmith
I use the mathbb to bold the i.
mathbb stands for 'math blackboard', not bold.
$$\mathbb{Z}$$ gives $\mathbb{Z}$

$$\boldmath{i}$$ gives $\boldmath{i}$

6. Originally Posted by dwsmith
I use the mathbb to bold the i.
If there is a problem using latex, common sense should be used to work around the problem. In this case, common sense dictates to just type i (which is what I did in my edit to the original post).

7. Originally Posted by Plato
mathbb stands for 'math blackboard', not bold.
$$\mathbb{Z}$$ gives $\mathbb{Z}$

$$\boldmath{i}$$ gives $\boldmath{i}$
My mistake \mathbf{} bolds the letters

8. Originally Posted by dwsmith
$\lim_{z\rightarrow -i}\frac{z^4-1}{z+ i}$

What do I do with a Complex limit when I get $\frac{0}{0}\mbox{?}$
You can also use L'Hospital's Rule.

$\displaystyle \lim_{z \to -i}\frac{z^4 - 1}{z + 1} = \lim_{z \to -i}\frac{\frac{d}{dz}(z^4 - 1)}{\frac{d}{dz}(z + 1)}$

$\displaystyle = \lim_{z \to -i}\frac{4z^3}{1}$

$\displaystyle = \lim_{z \to -i}{4z^3}$

$\displaystyle = 4(-i)^3$

$\displaystyle = 4i$.

This is also what you would get if you had factorised and cancelled the common factor of $\displaystyle z+i$.