Limits and Continuity in PreCalculus
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.