Limits and Continuity in Pre-Calculus - dummies

By Mary Jane Sterling

In mathematics, a limit suggests that you’re approaching some value. Some functions, such as a rational function with a horizontal asymptote, have a limit as the x values move toward positive or negative infinity — that is, as the value of x gets very small or very large. Limits are another way of describing the characteristics of particular functions.

Although limits are often demonstrated graphically (a picture is worth a thousand words?), you can describe limits more precisely using algebra.

Coupled with limits is the concept of continuity — whether a function is defined for all real numbers or not.

You’ll work on limits and continuity in the following ways:

  • Looking at graphs for information on a function’s limits

  • Using analytic techniques to investigate limits

  • Performing algebraic operations to solve for a function’s limits

  • Determining where a function is continuous

  • Searching for any removable discontinuities

When you’re working with limits and continuity, some challenges include the following:

  • Recognizing a function’s behavior at negative infinity or positive infinity

  • Using the correct technique for an analytic look at limits

  • Factoring correctly when investigating limits algebraically

  • Using the correct conjugates in algebraic procedures

  • Forgetting that the “removable” part of a removable discontinuity doesn’t really change a function’s continuity; a function with a removable discontinuity is not continuous

Practice problems

  1. Given the graph of f(x), find


    [Credit: Illustration by Thomson Digital]
    Credit: Illustration by Thomson Digital

    Answer: 3

    The function has a hole at (2, 3). The limit as x approaches 2 from the left is 3, and the limit as x approaches 2 from the right is 3.

  2. Determine the limit using the values given in the chart:


    [Credit: Illustration by Thomson Digital]
    Credit: Illustration by Thomson Digital

    Answer: ‒9

    The y values are getting closer and closer to ‒9 as x approaches ‒2 from the left and from the right.