Math 1210-001
Tuesday Jan 19
WEB L110
, Finish the "slope of the graph" limit example from Friday's notes.
, Section 1.5: Limits with infinity.
Example 1: Consider the rational function
f x =
3 xC1
.
xK2
1a) Compute
lim f x
x/0
1b) How can you compute
lim f x
as long as c is a real number with c s 2 ? (Hint, it's one of our limit theorems.)
x/c
1c) How about
lim
x/2
3 xC1
?
xK2
3 xC1
Below is (part of) a graph of y = f x =
to help visualize our discussion, as well as a table of
xK2
values of f for selected input x z 2, x s 2.
x
f x
1.9
K67
1.99
K697
1.999
K6997
2.1
73
2.01
703
2.001
7003
10
y
K4
K2
5
0
2
4
x
K5
K10
Def:
, arbitrarily large.
, xlim
/ cf
x =N means that for x close enough to c the values of f x can be made
lim f x =N means that for x close enough to c with x O c, the values of f x can be
x / cC
made arbitrarily large; lim f x =N means that for x close enough to c with x ! c, the values of f x can
x / cK
be made arbitrarily large;
, xlim
/ c f x =KN, lim f x =KN, lim f x =KN are defined analogously.
x / cC
x / cK
Def:
We
can also
think
about limits as the input variables x/N and as x/KN:
, lim
x /Nf x = L means that the values of f x get arbitrarily close to L as the inputs x
increase without bound.
, lim f x = L means that the values of f x get arbitrarily close to L as x decrease
x /KN
without bound.
1c) What is
lim
x /KN
x
3 xC1
?
xK2
f x
K10
2.417
K100
2.9314
K1000
2.9930
10
y
K20
K10
5
0
K5
K10
10
x
20
Exercise 2 Compute the following limits, if they exist. If the limits don't exist as real numbers, consider
the possibilities of GN.
4x
2a)
lim 2
x/1 x K9
4x
2b)
lim 2
x/3 x K9
4x
2c) xlim
.
/N 2
x K9
Summary: We can consider limits and one-sided limits of functions f x which are N or KN. These are
related to vertical asymptotes for the corresponding graphs. We can also take limits as x/N or x/KN.
These are related to horizontal asymptotes for the corresponding graphs for f.