Limits and Logarithms
Lecture Notes
page 1
Graphs of logarithmic functions:
If a > 1
f (x) = loga x
If 0 < a < 1
f (x) = loga x
y
-2
y
0
2
4
6
8
10
12
-2
0
2
4
6
8
10
12
x
x
Compute each of the following limits.
1. a) lim log2 x
x!1
b)
c) lim log2=3 x
e) lim ln x
d)
f)
x!1
lim log2 x
x! 1
2. a) lim log2 x5 + x2
x!1
b) lim log2
x5 + x2
c) lim log0:1
x5 + x2
d) lim ln 8x6
1
x!1
x!1
x!1
e) lim
lim log2=3 x
x! 1
j)
3 ln x
3 ln x
f) lim
x! 1 5 ln x + 2
4 (log2 x)2 7 (log2 x) + 3
g) lim
x!1
5 (log2 x) + 2
4 (log2 x)2 7 (log2 x) + 3
x! 1
5 (log2 x) + 2
lim
4 (log2 x)2 7 (log2 x) + 3
x!1
5 (log2 x)2 + 2
i) lim
4 (log2 x)2 7 (log2 x) + 3
x! 1
5 (log2 x)2 + 2
lim
lim
x! 1
ln x2
3
2
5 ln (x ) 1
log2 8x + log2 16x
x!1 log2 4x
log2 x
m) lim
n) lim
x!1
log2 8x + log2 16x
log2 4x + log2 x
12 + log3 27x
x!1 15 + log3 x
l) lim
4 log2 x3
x!1 5 + log2 (x7 )
o) lim
3. a) lim log2
12x2 3x + 1
3x2 + x + 1
d) lim log2
1
x
b) lim log2
40x2 x + 1
5x2 + 7x + 1
e) lim ln
3x2 + 1
x4 + 2
c) lim log2
40x2 x + 1
5x2 + 7x + 1
f) lim log2
1
x
x!1
x!1
x!1
c copyright Hidegkuti, 2013
lim ln x
x! 1
ln x + 1
k) lim p
x!1
ln x 1
l)
x!1 5 ln x + 2
h)
x!1
x!1
x!1
x!1
1
Last revised: October 15, 2013
Limits and Logarithms
Lecture Notes
page 2
Answers
1. a) 1
b) unde ned
c)
1
d) unde ned
2. a) 1
b) unde ned
c)
1
d) 1
h) unde ned
o)
3. a) 2
i)
4
5
j) unde ned
e)
3
5
e) 1
f) unde ned
f) unde ned
g) 1
n) 1
k) 1
l)
1
5
e)
1
f) unde ned
m) 1
l) 1
3
7
b) unde ned
c) 3
d)
1
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c copyright Hidegkuti, 2013
Last revised: October 15, 2013