Functions, Limits, Continuity: Limits

Functions, Limits, Continuity Limits

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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) = x² + 2 as x approaches 2 is 6:

Figure %: The Limit of f (x) = x² + 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

f (x) = 6 or

x²+2

= 6

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 piecewise-defined function:

f (x) =

Note that this function looks just like the function f (x) = x² + 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.

Two-Sided and One-Sided Limits

The standard limit that we've been talking about is a two-sided limit. It is considered two-sided 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)