Calculus
Exploration 2: Algebraic Limit Rules
Constant Multiple
Rule (CMR)
Sum of Functions
Rule (SR)
Difference of
Functions Rule (DR)
Product of
Functions Rule (PR)
Quotient of
Functions Rule (QR)
Constant Function
Rule (CFR)
Identity Function
Rule (IFR)
Power Function
Rule 1
(PFR 1)
Power Function
Rule 2 (PFR 2)
Power Function
Rule 3 (PFR 3)
Algebraic
Limit Rules: Example 1
Next, we
will carefully evaluate the following limit, using the limit rules
listed on the previous page, and justifying each step by labeling each
one with the appropriate rule(s) that is (are) used.
Algebraic
Limit Rules: Example 2
Again, we
will carefully evaluate the following limit, using the limit rules
listed on the previous page, and justifying each step by labeling each
one with the appropriate rule(s) that is (are) used.
Algebraic
Limit Rules: Example 3
In this
example, we will use the same function as in Example 2, but take the
limit as x approaches 1, instead. Notice that if we try to proceed as
in Example 2 we have:
If we try
to continue as in Example 2, we will end up with 0 in the denominator.
This illustrates that we can only apply the algebraic limit rules when
all of the component limits exist, and when we don't introduce division
by zero. So, we can't apply the Quotient Rule in this example. This
means we need a different approach to evaluate this limit (if it
exists). The key is to try a little algebra:
Notice that
this is an example of a function for which the limit exists as x
approaches 1, but 1 isn't in the domain of the function. What does the
graph of this function look like?
Algebraic
Limit Rules: Example 4
In this
example, we will use the reciprocal of the function in Example 3, and
take the limit as x approaches -1. In addition, we try to use the same
algebraic simplification, however in this case the limit does not
exist. What does the graph of this function look like?
Notice that
this function diverges in two different ways as x approaches -1 from
the left or from the right.
Algebraic
Limit Rules: Exercises
In each
exercise below, if possible, find the limit and list all the limit
rules used. Otherwise, explain why the limit does not exist.
Dr. John Pais,
Mathematics Department-MICDS
E-mail:
pais@micds.org or pais@kinetigram.com
URL:
http://kinetigram.com/micds