- Graph each piecewise defined function.
-
- Why is the following not a function?
Fundamentals
-
If
, then we say that is continuous at point . In general, there is a simpler way to think about continuity: if a section of a graph of a function can be drawn without lifting your pencil, then that part of the function is continuous. If you need to lift your pencil at some point, that point is called a discontinuity. If the whole function is continuous everywhere, then the function itself is called continuous.For each function below, label it as continuous or not continuous. If it is not continuous, list at least one discontinuity.
-
.
Continuousans
- The piecewise linear function
defined byDiscontinuous at .ans
- The piecewise linear function
defined byContinuousans
- The piecewise linear function
defined byDiscontinuity atans
- Your age in years as a whole number as a function of time.
Discontinuous at every birthdayans
- The height of a tennis ball as a function of time.
Continuousans
-
- In this exercise, we will compute
using a calculator.- Fill in the following table of values.
- What does
seem to equal?
ans
- Use a graphing calculator or computer to graph
. Looking at the graph, what does it look like is equal to when ?
Loooooks like .ans
- Fill in the following table of values.
- Compute the following limits using a calculator like in problem (4). In each case, sketch a graph and jot down a table of values.
-
ans
-
ans
-
- Watch the following KhanAcademy video link: One Sided Limits from Graphs
- Find the following values given the graph of
below.-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
DNE (or )ans
-
DNE (or )ans
-
DNEans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
ans
-
- From
from Problem 7, list whether is continuous or not continuous at the following values of : , , , , , .
Only continuous atans
- Watch the following KhanAcademy video link:
Two-sided limit from graph - Compute (as close as you can tell from the graph) the limit or function value in each case. If the limit does not exist, explain why.
-
ans
-
ans
-
Does not existans
-
1ans
-
ans
-
2ans
-
- Is it possible that
but ? If so, sketch a picture of the graph of such an If not, explain why not.
Yes, it is possible.ans
- Is it possible that
but If so, sketch a picture of the graph of such an If not, explain why not. - Suppose
.- What does
approach as approaches ?
ans
- What is
Unknownans
- What does
approach if is approaching ?
but also perhaps other valuesans
- What does
- Suppose
.- Estimate
.
It’s probably close toans
- Is is possible
? Why or why not?
This is possible but unlikely. There is nothing in the definition of limit that says can’t equal if the limit as is .ans
- Estimate
- For each part, sketch a graph of what
might look like. Each problem is separate and will probably require a different graph.-
and . -
does not exist. -
and . -
and .
-