CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 14. Concepts of Calculus

14.6 One Sided Limits and Continuity

Here you will determine one sided limits graphically, numerically and algebraically and use the concept of a one
sided limit to define continuity.
A one sided limit is exactly what you might expect; the limit of a function as it approaches a specificxvalue from
either the right side or the left side. One sided limits help to deal with the issue of a jump discontinuity and the two
sides not matching.
Is the following piecewise function continuous?

f(x) =







−x− 2 x< 1

− 3 x= 1

x^2 − 4 1 <x

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/62351

http://www.youtube.com/watch?v=3iZUK15aPE0 James Sousa: Determining Limits and One-Sided Limits Graph-
ically

Guidance

A one sided limit can be evaluated either from the left or from the right. Since left and right are not absolute
directions, a more precise way of thinking about direction is “from the negative side” or “from the positive side”.
The notation for these one sided limits is:

xlim→a−f(x),xlim→a+f(x)
The negative in the superscript ofais not an exponent. Instead it indicatesfrom the negative side. Likewise the
positive superscript is not an exponent, it just meansfrom the positive side. When evaluating one sided limits, it
does not matter what the function is doing at the actual point or what the function is doing on the other side of the
number. Your job is to determine what the height of the function should be using only evidence on one side.
You have defined continuity in the past as the ability to draw a function completely without lifting your pencil off of
the paper. You can now define a more rigorous definition of continuity.
If both of the one sided limits equal the value of the function at a given point, then the function is continuous at that
point. In other words, a function is continuous ataif:

xlim→a−f(x) =f(a) =xlim→a+f(x)