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Functions, Limits, Continuity

Math

Limits

Summary Limits
Figure %: The Limit of f (x) as x approaches c from the left


or "from the right" (i.e. from values of x greater than c):

Figure %: The Limit of f (x) as x approaches c from the right

However, not all functions behave like this. Some functions approach different values depending on whether we let x approach c from the left or from the right. For these functions, the two-sided limit does not exist, and we can only find the one-sided limit. Consider what happens to the following function as x approaches 3:

f (x) =    

Figure %: The behavior of f (x) as x approaches 3 from the left and from the right

As x approaches 3 from the left, f (x) approaches 9. We call 9 the left-hand limit of f (x) as x approaches 3, and we denote this as

f (x) = 9    

As x approaches 3 from the right, f (x) approaches 11. We call 11 the right- hand limit of f (x) as x approaches 3, and we denote this as

f (x) = 11    

Because there is no single value that f (x) approaches when x approaches 3, we must say that the standard two-sided limit, or f (x) does not exist. In general, f (x) exists only if f (x) = f (x) = L. In other words, the two-sided limit exists only if the left-hand and right-hand limits both exist and are equal.

Solving for Limits Using Limit Rules

Now that you know what limits are, you should become familiar with certain rules that allow you to manipulate and solve for them. Several of them should make intuitive sense.

Rule 1:
f (x) = f (c) if f (x) is a polynomial function. This means that if you are solving for the limit of a polynomial function at x = c, you can just plug x = c into the function to find the limit. For example,