Limits

Definition

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 limit of f(x) as x approaches x0 equals L

The entire discipline of calculus is based on the theory of limits. Let’s look at an example:

The limit as x approaches 1 of sqrt(x-1)/(x-1)

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:

Graph of f(x)=sqrt(x-1)/(x-1)

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:

Excel spreadsheet of f(x)=sqrt(x-1)/(x-1)

Based on our spreadsheet is appears that the limit as x→1 is 1/2.

Consider another example:

g(x)={3-x^2 for x<1; 1 for x=1; x+1 for x>1}

Graph of g(x) showing the removable discontinuity at x=1

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.

Limit from left:
The limit of f(x) as x approaches x0 from the negative side equals L.

Limit from right:
The limit of f(x) as x approaches x0 from the positive side equals L

Limit Theorems

The following theorems are quite handy when working with limits and particularly when writing calculus proofs involving limits.

First Limit Theorem

  1. For any constant C and any number x0

    The limit of C as x approaches x0 equals C

  2. If n is a positive integer or a positive fraction with an odd denominator, then

    The limit of x^n as x approaches x0 equals 0

    If n is a positive fraction with an even denominator, then

    The limit of x^n as x approaches 0 from the positive side equals 0

    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

    The limit of x^n as x approaches x0 equals x0^n

Second Limit Theorem

Second limit theorem.

Sandwich Theorem

Third Limit Theorem - the "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.


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