Let f(x) be a function. If f(x) approaches a number, L, as x approaches x0 then L is the limit of f(x) as x approaches x0. 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 x0 value.
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 x0
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 x0 is positive and n is any rational number, or if x0 is negative and n is either an integer or a fraction with an odd denominator, then
Second Limit Theorem
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.