Functions, Graphs, and Limits

Multiplication Property

As long as both and exist,

In words, the limit of a product is the product of the limits, as long as the limits involved exist.

Sample Problem

Find

If we try to break this limit into pieces, we find

This is trouble, because

What do we find if we multiply ∞ by something (if we could do such a thing, which we can't because ∞ is not a number)? It would make reasonable sense to assume we'd get ∞ again.

What do we get if we multiply a number by 0? We get 0. What if we try to multiply ∞ by 0? That makes no sense at all.

It turns out that in this case we can't find the limit of the product via the product of the limits.
However, we can still find the original limit we were asked about.

Since

=

Division Property

If the limits of f and g exist and

,

we find

Suppose and Then,

If , we can't use the rule we've been using to evaluate something like

If we try we find that

then we are out of luck.

Example 1

 Assume that \lim_x\to3g(x) exists, \lim_x\to3f(x) = 7, and \lim_x\to3(f(x)g(x)) = 28.Find \lim_x\to3g(x).

Example 2

 Assume that \lim_x\to2g(x) exists, \lim_x\to2f(x) = 7, and \lim_x\to2(xf(x)g(x)) = 28.Find \lim_x\to2g(x).

Example 3

 Assume that \lim_x\to1f(x) exists, \lim_x\to1g(x) = 5, and \lim_x\to1( \frac2f(x)g(x)) = 10.Find \lim_x\to1f(x).

Example 4

 Assume that \lim_x\to4f(x) exists and \lim_x\to4( \fracxf(x) + 203xf(x)) = 3.Find \lim_x\to4f(x).

Exercise 1

Find the following limit, assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32.

•  \lim_x\to10(f(x)g(x))

Exercise 2

Find the following limit, assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32.

•  \lim_x\to10(4xf(x)g(x))

Exercise 3

Find the following limit, assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32.

• \lim_x\to10(xf(x) + 20)

Exercise 4

Find the following limit, assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32.

• \lim_x\to10(x-g(x)\cdot g(x))

Exercise 5

Find the following limit, assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32.

\item \lim_x\to10(x2g(x) + f(x))

Exercise 6

Assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

• \lim_x\to10(\fracf(x)g(x))

Exercise 7

Assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

• \lim_x\to10(\frac1f(x))

Exercise 8

Assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

• \lim_x\to10(\fracf(x) + 2g(x)x-2)

Exercise 9

Assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

• \lim_x\to10(\fracx-3f(x)5)

Exercise 10

Assuming that\lim_x\to10f(x) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

• \lim_x\to10(\fracx-2f(x)g(x))

Exercise 11

Find \lim_x\to5f(x) (we may assume this limit exists).

•  \lim_x\to5(\fracf(x)x-2) = 2

Exercise 12

Find \lim_x\to5f(x) (we may assume this limit exists).

•  \lim_x\to5(2xf(x)) = 20

Exercise 13

Find \lim_x\to5f(x) (we may assume this limit exists).

• \lim_x\to5(f(x))2 = 16

Exercise 14

Find \lim_x\to5f(x) (we may assume this limit exists).

• \lim_x\to5(\frac14x-f(x)) = \frac12

Exercise 15

Find \lim_x\to5f(x) (we may assume this limit exists).

• \lim_x\to5(\frac\frac14x + f(x)4f(x)) = \frac32