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 limx→1– f(x)
Q3: Find limx→1+ f(x)
Q4: Find limx→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 limx→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 limx→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 limx→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 limx→1+ f(x) = 4
Q4: Find limx→1 f(x)
And finally, to find limx→1 f(x). limx→1 f(x) only exists if limx→1+ f(x) and limx→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,
limx→1 f(x) does not exist.