Audrey Markowskei
Department of Mathematics
Macquarie University
Sydney, Australia
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
y = 1/x
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
We write lim f (x) = 4 to say that the limit of f (x) as x
x!2
approaches 2 from the left is 4.
We write lim+ f (x) = 4 to say that the limit of f (x) as x
x!2
approaches 2 from the right is 4.
We write lim f (x) = 4. This means that f (x) gets close to 4
x!2
when x gets close to 2.
Definition
We write
lim f (x) = L
x!a
which means we can make the values of f (x) as arbitrarily
close to L as we like by taking x sufficiently close to a but not
equal to a.
Definition
lim f (x) = L
x!a
if and only if lim f (x) = L and lim+ f (x) = L, otherwise the
x!a
limit does not exist.
x!a
g(x) =
x 3 2x 2
x 2
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Find lim h(x)
x!0
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Find lim f (x) =
x! 2
3x+2
2x+4
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© 2012 Pearson Education, Inc.. All rights reserved.
YOUR TURN
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
x2 x 1
Find lim p
x!3
x +1
x2 x 1
Find lim p
x!3
x +1
x2 + x 6
x 2
x!2
Find lim
x2 + x 6
x 2
x!2
Find lim
Find lim
x!4
p
x 2
x 4
x 2 2x + 1
x!1 (x
1)3
Find lim
Limits at Infinity
For any positive real number n
1
lim n = 0, and
x!1 x
For any positive real number n
1
lim n = 0, and
x!1 x
1
= 0.
1 xn
lim
x!
1
= 0.
1 xn
lim
x!
Suppose a small pond normally contains 12 units of dissolved
oxygen in a fixed volume of water.
Suppose that at time t = 0 a quantity of organic waste is
introduced into the pond.
The oxygen concentration t weeks later is given by
12t 2
15t + 12
.
t2 + 1
What will be the oxygen concentration in the pond as time goes
on?
f (t) =
© 2012
Pearson
Education,
Inc.. AllInc..
rights
© 2012
Pearson
Education,
Allreserved.
rights reserved.
Limits at infinity (or negative infinity), if they exist, correspond to
horizontal asymptotes of the graph of a function.
lim
t!1
12t 2
15t + 12
t2 + 1
3x
x!1 4x 3
lim
3
1
3x 2 + 2
lim
x!1 4x
3