- Topics At a Glance
- Limits
- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
- All About Asymptotes
- Vertical Asymptotes
- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
**Manipulating Limits**- The Basic Properties
- Multiplication by a Constant
- Adding and Subtracting Limits
**Multiplying and Dividing Limits**- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.

- Assume

Then

- Assume

Then

- Assume

Then

Using the result from the previous example,

.

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

=

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\to3 Find \lim_x\to3g(x). |

Example 2

Assume that \lim_x\to2g(x) exists, \lim_x\to2 Find \lim_x\to2g(x). |

Example 3

Assume that \lim_x\to1 Find \lim_x\to1 |

Example 4

Assume that \lim_x\to4 Find \lim_x\to4 |

Exercise 1

Find the following limit, assuming that\lim_x\to10*f*(*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\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32.

- \lim_x\to10(4x
*f*(*x*)g(x))

Exercise 3

Find the following limit, assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32.

- \lim_x\to10(x
*f*(*x*) + 20)

Exercise 4

Find the following limit, assuming that\lim_x\to10*f*(*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\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32.

\item \lim_x\to10(x^{2}g(x) + *f*(*x*))

Exercise 6

Assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

- \lim_x\to10(\frac
*f*(*x*)g(x))

Exercise 7

Assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

- \lim_x\to10(\frac1
*f*(*x*))

Exercise 8

Assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

- \lim_x\to10(\frac
*f*(*x*) + 2g(x)x-2)

Exercise 9

Assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

- \lim_x\to10(\fracx-3
*f*(*x*)5)

Exercise 10

Assuming that\lim_x\to10*f*(*x*) = 5 and \lim_x\to10g(x) = \frac32,find the following limits.

- \lim_x\to10(\fracx-2
*f*(*x*)g(x))

Exercise 11

Find \lim_x\to5*f*(*x*) (we may assume this limit exists).

- \lim_x\to5(\frac
*f*(*x*)x-2) = 2

Exercise 12

Find \lim_x\to5*f*(*x*) (we may assume this limit exists).

- \lim_x\to5(2x
*f*(*x*)) = 20

Exercise 13

Find \lim_x\to5*f*(*x*) (we may assume this limit exists).

- \lim_x\to5(
*f*(*x*))^{2}= 16

Exercise 14

Find \lim_x\to5*f*(*x*) (we may assume this limit exists).

- \lim_x\to5(\frac14x-
*f*(*x*)) = \frac12

Exercise 15

Find \lim_x\to5*f*(*x*) (we may assume this limit exists).

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