Here is a brief example on the concept of a limit:

Look at the function f(x) in orange below:

Question 1.  Find f(3)

Explanation of question 1: Find the value of the function when you plug in 3.  What is the height of the function at the exact moment when x=3?

Answer 1:  The function is undefined at x=3.  There is a hole when x=3.

So, f(3) is undefined.

Question 2:  Find lim_x→3f(x)

Explanation of question 2:  We are being asked to find what the function is doing around (but not at) 3.  What is happening to the path of the function on either side of 3?

In order to find lim_x→3f(x), we must confirm that lim_x→3⁺  f(x) and lim_x→3^–  f(x) both exist and are equal to each other.

So, let’s find lim_x→3⁺  f(x).  What is happening to the function values as you approach x=3 from the right-hand side?  Literally run your finger along as if x=4, then x=3.5, then x=3.1.  What value is the function getting closer to?

The function is approaching a height of 4.

Let’s find lim_x→3^–  f(x).  What is happening to the function values as you approach x=3 from the left-hand side?  Literally run your finger along as if x=1, then x=2, then x=2.9.  What value is the function getting closer to?

The function is also approaching a height of 4.

So:

lim_x→3⁺  f(x) = 4

lim_x→3^–  f(x) = 4

Since, the left-handed limit at 3 and right-handed limit at 3 exist and are equal, this gives:

lim_x→3f(x) = 4.

So, to summarize, here are 4 different things we found.  They are related, but not necessarily the same:

f(3) is undefined

lim_x→3⁺  f(x) = 4

lim_x→3^–  f(x) = 4

lim_x→3f(x) = 4

Are you ready to try one on your own? Click here! (Don’t worry, I’ll walk you through the solutions too)