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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.2 • Limits of Functions Using Properties of Limits 91
y
A
f (x) 5 A
THEOREM The Limit of a Constant
If f (x) = A, where A is a constant, then for any real number c,
lim f (x) = lim A = A (1)
x→c x→c
The theorem is proved in Section 1.6. See Figure 17 and Table 8 for graphical and
numerical support of lim
x→c
A = A.
Teaching Tip
Depending on the function given, a
graphical approach may be helpful.
Consider verifying the limits using graphical
and analytical (that is, algebraic) methods.
Figure 17 For x close to c, the value
of f remains at A; lim x→c
A = A.
y
c
c
c
f (x) 5 x
Figure 18 For x close to c, the
value of f is just as close to c; lim x→c
x = c.
x
x
TABLE 8
x approaches c from the left
−−−−−−−−−−−−−−−−−−−−→
x approaches c from the right
←−−−−−−−−−−−−−−−−−−−−−
x c− 0.01 c − 0.001 c − 0.0001 → c ← c + 0.0001 c + 0.001 c + 0.01
f (x) = A A A A f(x) remains at A A A A
For example,
lim
x→5 2 = 2
lim 1
3 = 1 3
x→ √ 2
THEOREM The Limit of the Identity Function
If f (x) = x, then for any number c,
lim(−π) =−π
x→5
lim f (x) = lim x = c (2)
x→c x→c
This theorem is proved in Section 1.6. See Figure 18 and Table 9 for graphical and
numerical support of lim
x→c
x = c.
TABLE 9
x approaches c from the left
−−−−−−−−−−−−−−−−−−−−→
x approaches c from the right
←−−−−−−−−−−−−−−−−−−−−−
x c− 0.01 c − 0.001 c − 0.0001 → c ← c + 0.0001 c + 0.001 c + 0.01
f (x) = x c− 0.01 c − 0.001 c − 0.0001 f (x) approaches cc+ 0.0001 c + 0.001 c + 0.01
For example,
lim
x→−5 x =−5
lim
x→ √ x = √ 3
3
lim x = 0
x→0
1 Find the Limit of a Sum, a Difference, and a Product
Many functions are combinations of sums, differences, and products of a constant
function and the identity function. The following properties can be used to find the
limit of such functions.
Teaching Tip
Students do not have to be able to state
the theorems presented in this section.
However, they must be able to apply them.
Consider presenting these general rules:
• When finding the limit of a function, first
try to substitute the value of interest. If
that results in a finite value, or something
in the undefined form N for a nonzero N
0
(in which case the limit doesn’t exist), then
you’re done.
• If the substitution results in an
indeterminate expression (like 0 0 or ∞ ∞ ),
try to simplify the expression. If the
expression is rational, try to factor the
numerator and denominator and simplify.
If the rational expression contains a
binomial and one of the terms is a radical,
try multiplying by the conjugate.
• Remember, the ultimate goal is to
determine what value the function is
approaching as x gets very close to the
given value of interest.
IN WORDS The limit of the sum of two
functions equals the sum of their limits.
THEOREM Limit of a Sum
If f and g are functions for which lim f (x) and lim g(x) both exist,
x→c x→c
then lim[ f (x) + g(x)] exists and
x→c
A proof is given in Appendix B.
lim[ f (x) + g(x)] = lim f (x) + lim g(x)
x→c x→c x→c
TRM Section 1.2: Worksheet 1
This worksheet contains 6 questions for
finding limits by direct substitution. Graphs
are provided for answer verification.
Teaching Tip
As you get into the meat of this chapter, it may
be time to refer to the pacing guide and adjust
pacing. There are two schools of thought to
consider: (1) teaching to mastery level initially,
leaving only 3 weeks to review for the end-of-term
exam, or (2) moving more quickly through the
material, which then will leave 4 to 5 weeks for
review. The rationale for the second approach
is that many of the topics that students found
difficult at the beginning of the fall semester are
ones with which they are very comfortable by the
midterm in December. Here in Chapter 1, they still
don’t yet have the background and experience
to ask the good questions needed to push their
understanding of the concepts because they are
still focused on the nuts and bolts of getting the
correct answer.
Section 1.2 • Limits of Functions Using Properties of Limits
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