# Limit of a Function

### Definition of a Limit

Let f(x) be any function and let a and L be numbers. If we can make f(x) as close to L as we please by choosing x sufficiently close to a, then we say that the limit of f(x) as x approaches a is L or symbolically,

$\lim\limits_{x\to a}\left(f\left(x\right)\right)=L$

### Theorems on Limits

1. $\lim\limits_{x\to a}\left(c\right)=c$     ,   c=any constant
2. $\lim\limits_{x\to a}x=a$,      a=any real number
3. $\lim\limits_{x\to a}c\cdot f\left(x\right)=c\cdot \lim\limits_{x\to a}f\left(x\right)$
4. $\lim\limits_{x\to a}\left[f\left(x\right)+g\left(x\right)\right]=\lim\limits_{x\to a}f\left(x\right)+\lim\limits_{x\to \:a}g\left(x\right)$
5. $\lim\limits_{x\to a}\left[f\left(x\right)\cdot g\left(x\right)\right]=\lim\limits_{x\to a}f\left(x\right)\cdot \lim\limits_{x\to \:a}g\left(x\right)$
6. $\lim\limits_{x\to a}\:\frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim\limits_{x\to a}f\left(x\right)}{\lim\limits_{x\to \:a}g\left(x\right)}$
7. $\lim\limits_{x\to a}\:\sqrt[n]{f\left(x\right)}=\sqrt[n]{\lim\limits_{x\to a}f\left(x\right)}$     , n=any positive integer and f(x)>0 if n is even.
8. $\lim\limits_{x\to a}\left[f\left(x\right)\right]^n=\left[\lim\limits_{x\to \:a}f\left(x\right)\right]^n$

SOURCE: Differential and Integral Calculus by Feliciano and Uy