Limit of $\frac{sin(z)}{z }$ as $z$ approaches $0$?

Question

how can I proceed ? can I use the same techniques we use when computing a limit where the variable is a real number ?

edit : forgot to mention that z is a complex number

Answer 1

Hint: $$\lim_{z \rightarrow 0}\frac{\sin(z)-\sin(0)}{z-0}=(\sin)'(0)=\cos(0)=1$$

Answer 2

One technique is the use of power series:

For $ z \ne 0$: $ \frac{\sin z}{z}=1-\frac{z^2}{3!}+\frac{z^4}{5!}-+... \to 1$ for $z \to 0$

Answer 3

Maybe a more formal proof:

Denote $f(z)=\frac {\sin(z)}{z}$. $f$ is analytic in $\Bbb C$ except for a singularity point in $0$.

Look at the Taylor series $\sin(z)=\sum_{n=0}^\infty (-1)^n \frac {z^{2n+1}}{(2n+1)!}$. Which implies $\frac{\sin(z)}{z}=\sum_{n=0}^\infty (-1)^n \frac {z^{2n}}{(2n+1)!}$. This is still a Taylor series (there are no addends of the form $\frac{A}{z^{-n}}$ where $A\not= 0, n\in\Bbb N$). This means that $\lim_{z\to z_0}\frac{\sin(z)}{z}=\sum_{n=0}^\infty (-1)^n \frac {z^{2n}}{(2n+1)!}|_{z=0}=1$.

Remark: If you wouldn't have gotten a Taylor series the limit would have been $\infty$ if you would have gotten a series of the structure $\sum_{n=-m}^\infty a_{n}z^n$ and undefined if the series would have been of the structure $\sum_{n=-\infty}^\infty a_{n}z^n$ where $a_{-n}\not=0$ infinitely often for $n\in\Bbb N$.