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
Given the graph of f(x), find
Credit: Illustration by Thomson DigitalAnswer: 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.
Determine the limit using the values given in the chart:
Credit: Illustration by Thomson DigitalAnswer: ‒9
The y values are getting closer and closer to ‒9 as x approaches ‒2 from the left and from the right.