A one sided limit is exactly what you might expect; the limit of a function as it approaches a specific
Is the following piecewise function continuous?
Watch This
http://www.youtube.com/watch?v=3iZUK15aPE0 James Sousa: Determining Limits and OneSided Limits Graphically
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:
The negative in the superscript of
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 at
Example A
What are the one sided limits at 5, 1, 3 and 5?
Solution:
Example B
Evaluate the one sided limit at 4 from the negative direction numerically.
Solution: When creating the table, only use values that are smaller than 4.

3.9  3.99  3.999 

0.9  0.99  0.999 
Example C
Evaluate the following limits.

limx→3−(4x−3) 
limx→2+(1x−2) 
limx→1+(x2+2x−3x−1)
Solution: Most of the time one sided limits are the same as the corresponding two sided limit. The exceptions are when there are jump discontinuities, which normally only happen with piecewise functions, and infinite discontinuities, which normally only happen with rational functions.
a.
b.
The reason why
c.
Concept Problem Revisited
In order to confirm or deny that the function is continuous, graphical tools are not accurate enough. Sometimes jump discontinuities can be off by such a small amount that the pixels on the display of your calculator will not display a difference. Your calculator will certainly not display removable discontinuities.
You should note that on the graph, everything to the left of 1 is continuous because it is just a line. Next you should note that everything to the right of 1 is also continuous for the same reason. The only point to check is at

limx→a−f(x)=−1−2=−3 
f(1)=−3 
limx→1+f(x)=12−4=−3
Therefore,
Vocabulary
A one sided limit is a limit of a function when the evidence from only the positive or only the negative side is used to evaluate the limit.
Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For an entire function to be continuous, the function must be continuous at every single point in an unbroken domain.
Guided Practice
1. Megan argues that according to the definition of continuity, the following function is continuous. She says

limx→2−f(x)=∞ 
limx→2+f(x)=∞ 
f(2)=∞
Thus since
2. Evaluate the following limits.

limx→1−(2x−1) 
limx→−3+(2x+2) 
limx→2+(x3−8x−2)
3. Is the following function continuous?
Answers:
1. Megan is being extremely liberal with the idea of “
2.

limx→1−(2x−1)=2⋅1−1=2−1=1 
limx→−3+(2x+2)=2−3+2=2−1=−2 
limx→2+(x3−8x−2)=limx→2+((x−2)(x2+2x+4)(x−2))=limx→2+(x2+2x+4)=22+2⋅2+4=12
3. Use the definition of continuity.

limx→1−f(x)=(−1)2−1=1−1=0 
f(−1)=3 
limx→−1+f(x)=−1+3=2
Practice
Evaluate the following limits.
1.
2.
3.
4.
5.
6.
Consider
7. What is
8. What is
9. Is
Consider
10. What is
11. What is
12. Is
Consider
13. What is
14. What is
15. Is