In Calculus, a limit is used to describe the value that a function approaches as the input of that function approaches a certain value. There are two ways to determine a Calculus limit: a numerical approach or a graphical approach. In the graphical approach, we analyze the graph of the function to determine the points that each of the one-sided limits approach.
I want to talk about one sided limits, here's a function g of x equals and it's piece y is defined x+8 for x less than -4 and x squared -1 for x greater than or equal to -4. Describe its behavior as x approaches -4 well at -4 that's where the two pieces are kind of joined together and so it may behave differently depending on what side of -4 we're on. So let's try approaching -4 from the left -5 is to the left of -4 on the number line so we're starting from the left and moving to the right. Uh -5, -4.1, -4.01 look what happens to the values we get 3, 3.9, 3.99 these values are getting closer and closer to 4. When x is less than -4 we're using this piece of the function so we're getting closer and closer to the value 4. Now if we start from the right like -3 is to the right of -4 and I go to the left I get -3, -3.9, -3.99. These are the values I get 8, 14.2, 14.9 and you can see that these values seem to be getting closer and closer to 15. Okay so we're approaching 4 from the left and approaching 15 from the right. So here's what we say, we say the limit as x approaches -4 from the left of f of x sorry g of x this is g is 4. This is the left hand limit, the left hand limit is one of the one sided limits for g of x at 4. And then we say limit as x approaches -4 from the right of g of x is 15. This is the right hand limit, this little uh superscript tells you which it is whether it's the left hand or the right hand limit. The superscript negative means that you're approaching -4 from the left from the more negative direction. And the superscript plus means you're approaching -4 from the right from the more positive direction. Now notice these two limits are not equal, whenever the two one sided limits are not equal the two sided limit x approaches -4 in this case does not exist very important. So in order for a two sided limit like this to exist you need both of the one sided limits to exist and for them to be equal. And that's what this theorem states right here. Limit as x approaches a of f of x equals l if and only if the two one sided limits the limit is x approaches a from the left and the limit is x approaches a from the right of f of x equal l the same number. Only if those two one sided limits have the same value will the two sided limit exist.