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