Calculus: Intuitive look at a limit

A limit looks at what happens to a function when the input approaches a certain value. The general notation:

Let's say that the function that we're interested in is f(x) = x², and that we're interested in its limit as x approaches 2. Using the above notation:

One way to try to evaluate what this limit is would be to choose values near 2, compute f(x) for each, and see what happens as they get closer to 2:

One way to try to evaluate what this limit is would be to choose values near 2, compute f(x) for each, and see what happens as they get closer to 2. This is implemented as follows:

x	1.7	1.8	1.9	1.95	1.99	1.999
f(x) = x2	2.89	3.24	3.61	3.8025	3.9601	3.996001

Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above:

x	2.3	2.2	2.1	2.05	2.01	2.001
f(x) = x2	5.29	4.84	4.41	4.2025	4.0401	4.004001

We can see from the tables that as x grows closer and closer to 2, f(x) seems to get closer and closer to 4, regardless of whether x approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of x² as x approaches 2 is 4.