Limits Part I

Calculus is based on the idea of a limit. We are not going to carefully define limits in terms of epsilon and delta right now, but we are just going to practice taking some limits of some functions.

The limit of a function, in roundabout terms, is the value of as approaches some number, say c. When we take the limit of a function we are not interested in what the value of the function is at c, rather we are interested in the values of the function around c.

Suppose we have a function, which is graphed below, and we want to find the limit of as approaches 3. We can look at the graph of to see how the functions behaves around c = 3.

As you can see(may be not so obviously), is defined by the equation, at all points except = 3. At = 3, = -5. However, we are not interested at how behaves at 3. Around = 3, the functions approaches 10, both from the right and the left. So the limit of as approaches 3, is equal to 10. Mathematically we can write this statement as:

There is one last topic on the introduction of limits that needs to be discussed and that is the direction of the limit. In the above graph, the limits approached 10 from both the right and the left. In mathematics, we can be more specific on where the limit is coming from. Let's take a look at the following graph to help determine the limit from the left and the limit from the right:

This limit is broken at x = -2. The limit of as x approaches -2 from the left is -10, and the limit of as x approaches -2 from the right is 8. We can express this the following way:


The superscripted negative and positive tell which direction we are taking the limit of from.