SECTION 1.3:
EVALUATING LIMITS ANALYTICALLY
Goals: The Student Will Be Able To:
Recall, understand and apply basic limit properties and theorems
Recognize and use direct substitution when appropriate
Understand and apply the theorem of functions that agree in all but one point
Use the cancellation technique to eliminate factors in rational functions
Use the rationalization technique for functions involving radicals in the numerator or denominator
Understand and explain The Squeeze Theorem
Recognize and apply the two special trigonometric limits
PROPERTIES OF LIMITS
Direct Substitution
The basic idea is that simple direct substitution of the “c” value into the function can be used to evaluate a limit.
When is this permitted?
The function must be “well-behaved” … that is to say “continuous at c” … We will explore various functions that you have had vast experiences with in past courses … but first …
Theorem 1.1: Some Basic Limits
1. The limit of a constant # is that #
2. As x approaches c, the expression x approaches c … it is almost redundant
3. For any integer power of x, you can just plug in the c value
EXAMPLE 1:
1. 2.
3.
Theorem 1.2: Properties of Limits
Assume: b and c are real numbers, n is a positive integer, and f and g are functions with the following limits:
1. Scalar Multiple:
2. Sum or Difference:
3. Product:
4. Quotient:
5. Power:
EXAMPLE 2:
Find the limit of the polynomial:
1.2: Property 2
1.2: Property 1
1.1: Property 3 & 1
Simplify
Theorem 1.3: Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
Direct substitution is always permitted to find a limit of a polynomial function
If r is a rational function given by r(x)=p(x)/q(x) and c is a real number such that q(c) 0, then
Direct substitution is always permitted to find a limit of a rational function, provided the denominator is not 0
EXAMPLE 3:
Find the limit:
Solution: Since the value of the denominators is not 0, when plugging in 1, we can perform direct substitution to evaluate the limit.
Theorem 1.4: The Limits of a Function Involving a Rdaical
Let n be a positive integer. The following is valid for all c if n is odd, and is valid for c > 0 if n is even.
Direct substitution is always permitted to find a limit of a radical function provided the radicand is not negative for an even index
Theorem 1.5: The Limit of a Composite Function
If f and g are functions such that:
and ,
then
Direct substitution is always permitted to find a limit of a composite function
EXAMPLE 4:
Find the limit of the composite functions:
a.
b.
Theorem 1.6: Limits of Trigonometric Functions
Let c be a real number in the domain of the given function:
1. 2.
3. 4.
5. 6.
EXAMPLE 5:
Find the trigonometric limit:
a.
b.
c.
Theorem 1.7: Functions That Agree at All But One Point
Let c be a real number and let for all in an open interval containing c. If the exists, then also exists, and
EXAMPLE 6:Find the limit:
First, note why direct substitution will not work!!!
… but there is a related function … identical to this one … except for at x = 1.
So, to evaluate the original limit for f, we will consider it for g instead:
Graphically, we can explore the similarity and difference in the 2 functions … let’s look at the graphing calculator…
CANCELLATION AND RATIONALIZATION TECHNIQUES
This last example used algebraic factoring, and the canceling of common numerator and denominator factors to create a new g(x) function. The limit of this new g(x) function was easily found by direct substitution.
EXAMPLE 7:Find the limit:
Again, note why direct substitution fails:
Factor:Cancel:
Substitute: … Now, reference the graphs.
Incidentally:
We have seen the fraction appear in both of the last 2 examples. This form is NOT called “UNDEFINED” … it IS called an “INDETERMINATE FORM”. The reason for this is that the limit cannot be determined while the expression is in that form.
EXAMPLE 8:Find the limit:
Attempt direct substitution first:
What form do we again have here???
Now for something that you may not like! We will rationalize the numerator!!?? Yes … the numerator!!!
To do so, we must multiply numerator and denominator by: THE CONJUGATE of the radical expression.
… Now we can again attempt the limit.
We can confirm this both numerically and graphically.
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