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→3}f(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→3}f(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→3}f(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→3}f(x) = 4

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

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