1
c Amy Austin, September 4, 2012
Section 2.2: Limits
sin(x)
. This function is not defined at
The concept of a limit Consider f (x) =
x
x = 0. However, we can get as close to 0 as we wish, provided we never reach 0.
As x gets closer and closer to 0, we want to know if f (x) approaches what we call a
limiting value. To investigate this, we will evaluate f (x) for x close to 0.
x →0−
−1
−0.5
−0.1
−0.05
−0.01
−0.001
−0.0001
f (x)
0.8414709848
0.9588510772
0.9983341665
0.9995833854
0.9999833334
0.9999998333
0.9999999983
x →0+
1
0.5
0.1
0.05
0.01
0.001
0.0001
f (x)
0.8414709848
0.9588510772
0.9983341665
0.9995833854
0.9999833334
0.9999998333
0.9999999983
By viewing the table and graph above, it appears as x → 0− ,
sin(x)
→ 1, and
x
sin(x)
sin(x)
→ 1. Therefore
approaches 1 as x approaches 0
x
x
sin(x)
equals 1,
from both directions. We then say the limit as x approaches 0 of
x
sin(x)
= 1.
and we write lim
x→0
x
likewise, as x → 0+ ,
Definition:
(i) If lim− f (x) = L and lim+ f (x) = L, then lim f (x) exixts and lim f (x) = L.
x→a
x→a
x→a
x→a
(ii) If lim− f (x) 6= lim+ f (x), then lim f (x) does not exist (DNE).
x→a
x→a
x→a
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c Amy Austin, September 4, 2012
Graphical Limits
7
lim f (x) =
x→−1−
f(x)
, lim f (x) =
lim f (x) =
lim f (x) =
1
0
0
1
0
1
f (−5)
5
, lim f (x) =
,
x→1
, f (−1)
lim f (x) =
x→−3+
,
x→1
x→−5
11
00
00
11
,
x→−1
x→−1
−5
, lim + f (x) =
f (−3)
, lim − f (x) =
x→−3
1
0
0
1
0
1
−7
Limits of piece-wise defined functions

2 − x if x < −1



x
if −1 ≤ x < 1
EXAMPLE 1 :f (x) =

if x = 1

4
4 − x if x > 1
Sketch the graph of f (x) and find all values of x for which the limit does not exist.
,
3
c Amy Austin, September 4, 2012
Infinite Limits and vertical asymptotes: If lim+ f (x) = ±∞ or
x→a
lim− f (x) = ±∞, then we say x = a is a vertical asymptote of f (x).
x→a
1
x2
xx →0+
1
0.5
0.1
0.05
0.01
0.001
0.0001
Infinite Limits: Calculate lim
x→0
f (x)
x →0−
−1
1
−0.5
4
−0.1
100
−0.05
400
−0.01
10, 000
−0.001
1, 000, 000
−0.0001 100, 000, 000
f (x)
1
4
100
400
10, 000
1, 000, 000
100, 000, 000
Discussion: What characteristics must f (x) have in order to contain vertical asymptotes in its graph?
EXAMPLE 2: Find the infinite limit:
6
(a) lim−
x→5 x − 5
(b) lim+
x→5
(c) lim
x→5
6
x−5
6
x−5
4
c Amy Austin, September 4, 2012
(d) lim+
x−1
x2 (x + 2)
(e) lim−
x−1
x2 (x + 2)
x→0
x→0
x−1
x→0 x2 (x + 2)
(f) lim
(g) lim− csc x
x→π
x+1
. For each vertical
− 2x − 3
asymptote, describe the behavior of f (x) near the asymptote.
EXAMPLE 3: Find all vertical asymptotes for f (x) =
x2