What is the importance of uniqueness theorem of a limit? Almost in every calculus book the uniqueness theorem of a limit is the first theorem. Which can be stated as follows

If $\lim_{x\to a} f(x)=L_1$ and $\lim_{x\to a} f(x)=L_2$, then $L_1=L_2$.

What is the importance of this theorem? 
 A: If that theorem were false, then you couldn't coherently talk about limits at all, and analysis would look very, very different. This is one of the many theorems of the form "the thing which has to be true is, in fact, true".
A: Before you have proved that limits are unique, even writing “$\lim_{x\to a} f(x) = L$” is meaningless.
To be pedantic, one should first define what “$f(x)\to L$ as $x \to a$” means, and then formulate the theorem as follows:

If $f(x)\to L_1$ as $x \to a$ and $f(x)\to L_2$ as $x \to a$, then $L_1=L_2$.

Once you know this (but not before), it makes sense to introduce $\lim_{x\to a}f(x)$ as notation for the unique number $L$ such that $f(x) \to L$ as $x \to a$.
A: As mentioned by the others, the theorem is very fundamental!
Especially because it is just true for the setting you have, I assume $f$ is a map between metrical spaces?
For metrical spaces or more structured space, the limit is unique and everything works fine. But you can define continuity and convergence for sequences even for topological spaces and in topological spaces, a limit has not to be unique! And then a lot of arguments don't work anymore.
