Section 1.3: Evaluating Limits Analytically
Properties of Limits:
The limit of f(x) as x approaches c does not depend on the value of f at x = c. However, it may
happen that the limit is precisely f(c). In such cases, we can evaluate the limit by direct substitution.
𝐥𝐢𝐦 𝒇(𝒙) = 𝒇(𝒄)
𝒙→𝒄
Substitute c for x
Theorem 1.1: Some Basic Limits:
Let b and c be real numbers, and let n be a positive integer.
1. lim 𝑏 = 𝑏
%→8
2. lim 𝑥 = 𝑐
%→8
3. lim 𝑥 ; = 𝑐 ;
%→8
4.
5. lim 𝑏 = 𝑏
Example 1: Use the Direct Substitution Property of Limits to evaluate the following limits
a. lim 4 =
%→'
b.
lim 𝑥 =
%→*+
c. lim 𝑥 - =
%→'
d. lim 𝑥 ' =
%→*-
1
Theorem 1.2: Properties of Limits
Example 2: Find the limit: lim (4𝑥 ' + 3)
%→'
Note: the limit (as x approaches 2) of the polynomial function 𝑝 𝑥 = 4𝑥 ' + 3 is the value of p at x
= 2.
lim(4𝑥 ' + 3) = 4(22) + 3 = 19
%→'
è This direct substitution property is valid for all polynomial and rational functions with nonzero
denominators.
▪
Example of polynomial functions: 𝑝 𝑥 = 4𝑥 ' + 3; 𝑝 𝑥 = 5𝑥 - + 𝑥 ' − 7
▪
Example of rational functions with nonzero denominators: 𝑟 𝑥 =
% C D -%
+% D E
;𝑟 𝑥 =
'% D -
2
Theorem 1.3: Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
lim 𝑝(𝑥 ) = 𝑝(𝑐)
Direct substitution
%→8
I(%)
If r is a rational function given by 𝑟(𝑥 ) = J(%) and c is a real number such that
𝑞(𝑐 ) ≠ 0, then
I(8)
lim 𝑟(𝑥 ) = 𝑟(𝑐 ) = J(8)
%→8
Direct substitution
Example 3: Find the limit
a.
lim (2𝑥 ' + 4𝑥 + 1)
%→*-
b. lim
%
%→E % C D '
Theorem 1.5: The Limit of a Composite Function
Example 4: Find the limit
lim 𝑥 ' + 4 =
%→H
3
We have seen the limits of many algebraic functions can be evaluated by direct substitution. The six
basic trigonometric functions also exhibit the same quality, as shown in the below theorem
Example 5:
a. lim sin (𝑥) =
%→H
b. lim (𝑥 𝑐𝑜𝑠𝑥) =
%→Q
You have studied several types of functions whose limits can be evaluated by direct substitution.
This knowledge, together with the next theorem, can be used to develop a strategy for finding
limits.
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Example 6: Find the limit:
𝑥- − 1
lim
%→E 𝑥 − 1
You can see the two graphs below of the functions f(x) and g(x), they agree everywhere except at
the target point x = 1.
A Strategy for Finding Limits:
5
Dividing Out Technique:
Another procedure for finding a limit analytically is the dividing out technique. This technique
involves diving out common factors.
H
In some cases, direct substitution produced the meaningless fractional form which is called an
H
indeterminate form because you cannot determine the limit directly.
è Solution: you must rewrite the fraction so that the new denominator does not have 0 as its
limit. One way to do this is to divide out common factors.
Example 7:
1. Find the limit: lim
%→*-
2.
lim
%→*U
3. lim
%→'
% C * ET
%DU
% C D %*T
%D-
=
% C DV%*EW
% C *U
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Rationalizing Technique:
Another way to find a limit analytically is the rationalizing technique, which involves
rationalizing either numerator or denominator of a fractional expression.
We know that rationalizing the numerator (denominator) means multiplying the numerator and
denominator by the conjugate of the numerator (denominator).
Example 8:
1. lim
%→-
%DE*'
%*-
7
2. lim
%→U
%D+*%*U
8
The Squeeze Theorem:
The next theorem concerns the limit of a function that is squeezed between two other functions,
each of which has the same limit at a given x-value.
Example 9:
If 10𝑥 ≤ 𝑓 𝑥 ≤ 6𝑥 ' + 4 for 0 ≤ x ≤ 2, evaluate:
lim 𝑓(𝑥) =
%→E
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