Limits
Definition
Let f(x) be a function. If f(x) approaches a number, L, as x approaches x_{0} then L is the limit of f(x) as x approaches x_{0}. This is generally notated as:
The entire discipline of calculus is based on the theory of limits. Let’s look at an example:
By inspection, we can see that this function is undefined at x = 1 because the bottom of the rational expression evaluates to 0 at this point. We can easily see this discontinuity when we graph the function:
Even though the function is undefined at x = 1, however, the limit still exists. We could approximate it fairly easily using an Excel spreadsheet, or using a calculator:
Based on our spreadsheet is appears that the limit as x→1 is 1/2.
Consider another example:
This example is a piecewise function with a removable discontinuity at x = 1. Note that although the value of g(1) = 1, the limit of g(x) as x→1 is actually 2.
Finally, please be aware that a function can have different limits at a point depending on whether you are coming from the left or the right. This is notated by a positive or negative sign next to the x_{0} value.
Limit Theorems
The following theorems are quite handy when working with limits and particularly when writing calculus proofs involving limits.
First Limit Theorem

For any constant C and any number x_{0}

If n is a positive integer or a positive fraction with an odd denominator, then
If n is a positive fraction with an even denominator, then
If x_{0} is positive and n is any rational number, or if x_{0} is negative and n is either an integer or a fraction with an odd denominator, then
Second Limit Theorem
Sandwich Theorem
Sources
Shenk, A. (1988). Calculus and Analytic Geometry [4th Ed.]. Glenview, IL: Scott Foresman and Company.
Swokowski, E., Olinick, M., Pence, D. & Cole, J. (1994). Calculus of a Single Variable [6th Ed.]. Boston, MA: PWS.