Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
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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.

About This Article

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About the book author:

Mary Jane Sterling is the author of Algebra I For Dummies, Algebra Workbook For Dummies, and many other For Dummies books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics.

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