Visualizing limits vs values

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.


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