Function Limits I: The Intuition

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Lesson Summary: 
Function limits show how close a function can get to a point on the y-axis as it approaches a number on the x-axis. Here we introduce notation and begin developing intuition through pictures.
Lesson Inputs: 
Functions
Lesson Outputs: 
Function Limits II: When Function Limits Don't Exist

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Lesson: 

Function limits explain in mathematical detail how close a function gets to some number on the y-axis as it approaches a number on the x-axis. We oftentimes refer to the number on the y-axis as the function's limit and denote it by $L$. The function limit answers the question: "As $x$ approaches $a$, what does $f(x)$ intend to be?"

A graph of the basic idea behind a function limit.

In the picture above, as $x$ approaches $a$, the function approaches the value $L$. In English we would say "The limit of $f(x)$ as $x$ approaches $a$ is $L$." In math we would use the notation

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It is important for you to get used to the limit notation. Differential Calculus will use function limits a lot. You could even say that Differential Calculus is the study of function limits as you will see when you begin studying derivatives.

As the function gets closer and closer to the point $a$, the function approaches a value $L$. Notice that I did not say the function approaches the value $f(a)$. The value of the function at $a$ does not have to be the same as its limit at $a$! This is an extremely important point to remember. The two pictures below demonstrate this point. In the first one we have a discontinuity at $a$ and in the second the functions does not exist at all at $a$.

A graph of a function limit where the function has a discontinuity.

So what type of function would look like this? One example would be

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This function is just $x^2$ with a discontinuity at $x=3$. Normally the function would be 9 at $x=3$, but this function has the value of 1 at $x=3$.

A graph of a function limit where the function has a hole.

Here is an example of a function with a hole in it.

If the limit at a hole and discontinuity confuses you then don't worry! I just want you to keep the pictures in your head and accept it for now. In the lesson "Function Limits III: The Gory Details" we will provide a mathematical definition of function limits and prove that the function limit can still exist even though the function has a hole or discontinuity in it. The next step is for you to understand when function limits don't exist.