Limits: an Intuitive Definition
Intuitively, the limit of f (x) as x approaches c is the value
that f (x) approaches as x approaches c. For example, the limit of f (x) = x^{2} + 2
as x approaches 2 is 6:
Figure %: The Limit of f (x) = x^{2} + 2 as x approaches 2
As x gets closer and closer to 2, f (x) gets closer and closer to 6. In mathematical
notation, we can represent this as
Note that we've only been talking about what happens to f (x) as xapproachesc, and not about what happens when xequalsc. The
truth is that when we're looking for limits, we don't care what happens to f (x) when
x actually equals c  we're only concerned with its behavior as x gets closer and
closer to c. Consider the following piecewisedefined function:
f (x) = 

Note that this function looks just like the function f (x) = x^{2} + 2, except that f (2) = 9
instead of 6. What happens when we try to find
f (x) ? 

Figure %: The Limit of f (x) as x approaches 2
We see that the limit is again 6. Once again, this is because the limit doesn't care
what happens when x = c! As long as two functions approach the same value as
x approaches c, their limits will be the same.
TwoSided and OneSided Limits
The standard limit that we've been talking about is a twosided limit. It is
considered twosided because we get the same value for the limit whether we let x
approach c "from the left" (i.e. from values of x less than c)