2.5 One-Sided Limits and Continuity

One-Sided Limits

The function f has the right-hand limit L as x approaches a from the right, written

The function f has the left-hand limit L as x approaches a from the right, written

A limit can exist if and only if the one-sided limits agree.

Find the limit, the right-hand limit, and the left-hand limit from this graph.

Limit =

Right-hand limit =

Left-hand limit =

A Function is Continuous if it satisfies all of the following
1) f(a) exists.

2) a limit, , must exist

3)

*Remember that a function is continuous if you can trace the entire graph with your finger

Continuous Functions
1) Polynomials
2) Constants
3) Sin and Cos
4) Exponential
5) Logarithmic (on their domain)

Is the function on the graph continuous?

No, because the function is not continuous at x = -2

Intermediate Value Theorem

If f(x) is a continuous function on a closed interval [a, b] and M is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) =M

Existence of Zeros of a Continuous Function

If f is a continuous function on a closed interval [a, b], and if f(a) and f(b) have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval [a, b]

Example Problem

Calculate the left and right hand limits of f (x) at 2.

As x approaches 2 from the left, f (x) = 3 - x and f(x) = 1 at x=2 so the left-hand limit is 1

As x approaches 2 from the right f(x) = x/2 and f(x) = 1 at x=2 so the right hand limit is 1.

Here are some helpful links:

http://tutorial.math.lamar.edu/AllBrowsers/2413/Continuity.asp

http://tutorial.math.lamar.edu/AllBrowsers/2413/OneSidedLimits.asp

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