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

We are going to answer 4 questions about this graph. They are all related to each other, but different questions. Seeing the difference will help us sort out the difference between a function value and a limit.

**Q1: Find f(1)**

**Q2: Find lim _{x→1–} f(x)**

**Q3: Find lim _{x→1+} f(x)**

**Q4: Find lim _{x→1} f(x)**

OK….. Try to answer these questions with what you know… Then continue reading to see the answers and explanations!

**Q1: Find f(1)**

This question is asking you to find the value of f(x) when x=1. So, when you plug 1 into the function, where is the “point” located? You can look vertically above where x=1 and see that there is a solid point at height 4 (and a hole at 2). The value of the function exists at the solid point.

So, f(1) = 4.

**Q2: Find lim _{x→1–} f(x)**

This question is asking you to find the limit of f(x) as x approaches 1 from the left-hand side. As we move closer to 1 from the left hand side, the function approaches a height of 2.

So, **Find lim _{x→1–} f(x) = 2**

*Remember, we don’t care that there is actually a hole sitting there! The question is about the limit — where is the path going (we don’t care what happens when you get there, just where are you going).*

**Q3: Find lim _{x→1+} f(x)**

Same game, but now coming from the right-hand side. Follow the function and approach 1 from the right-sided path. As you get closer to 1, the function approaches 4.

So, **Find lim _{x→1+} f(x) = 4**

**Q4: Find lim _{x→1} f(x)**

And finally, to find lim_{x→1} f(x). lim_{x→1} f(x) only exists if lim_{x→1+} f(x) and lim_{x→1–} f(x) exist and are equal. Well, the left-handed limit and the right-handed limit at x=1 both exist, but they are not equal. So,

lim_{x→1} f(x) does not exist.