# Functions, Graphs, and Limits

## Introduction to Functions, Graphs, And Limits - At A Glance:

Limits can be added and subtracted, but only when those limits exist.

## Adding and Subtracting Property 1

If a is a real number and both

and

exist, then

In words, as long as the limits that are added both exist, "the limit of the sum is the sum of the limits.''

= 3 + 9 = 12.

### Sample Problem

By the previous rule about "pulling out'' constants,
= 4(3) = 12.

Therefore,

= 12 + 9 = 21.

Often we use this rule in reverse. Instead of evaluating

by breaking it up into two limits to evaluate, we would evaluate

by combining the limits into one.

This stuff works for subtraction, too.

## Adding and Subtracting Property 2

If a is a real number and both

and

exist, then

In words, as long as the limits that are subtracted exist, "the limit of the difference is the difference of the limits.''

### Sample Problem

Now, here's a non-example to show why we need each of the limits that are added or subtracted to exist.

### Sample Problem

Since

does not exist, we are not allowed to say

=

We can't evaluate the left-hand side of that equation, but we can evaluate the right-hand side.

If we want to evaluate

we're out of luck.

does not exist,

and we can't subtract "does not exist'' from "does not exist,'' since that doesn't make sense.

Infinity isn't a number, so "∞-∞"' doesn't make sense.

The point is that we can't perform the subtraction

,

since evaluating the limits doesn't give us actual numbers to subtract.

Now for the right-hand side. Since

\frac1x-\frac1x = 0,

we can easily evaluate this limit:

The addition and subtraction rules also extend to adding several functions. as long as the limits

, ,

all exist,

#### Example 1

 Given that \lim_x\to2f(x) = 1,find \lim_x\to23f(x) + 8x-4.

#### Example 2

 Assuming the limit exists, find \lim_x\to3f(x)given that \lim_x\to3(f(x) + 5) = 14.

#### Example 3

 Find \lim_x\to5f(x),given that this limit exists and \lim_x\to5(x-f(x)) = 12.

#### Example 4

 Assuming that \lim_x\to 10f(x) exists and that \lim_x\to 10(4x + 2-3f(x)) = 21, find \lim_x\to 10f(x).

#### Exercise 1

Given that

\lim_x\to12f(x) = 7,

evaluate the following limit.

#### Exercise 2

Given that

\lim_x\to12f(x) = 7,

evaluate the following limit.

#### Exercise 3

Given that

\lim_x\to12f(x) = 7,

evaluate the following limit.

\lim_x\to12(10-2f(x))

#### Exercise 4

Given that

\lim_x\to12f(x) = 7,

evaluate the following limit.

\lim_x\to12(f(x) + 3x-1)

#### Exercise 5

Given that

\lim_x\to12f(x) = 7,

evaluate the following limit.

\lim_x\to1210f(x)-\lim_x\to12(3 + 5f(x))

#### Exercise 6

Assuming that \lim_x\to4f(x) exists, find this limit if

\lim_x\to420f(x) = 100

#### Exercise 7

Assuming that \lim_x\to4f(x) exists, find this limit if

\lim_x\to4(5f(x) + 11) = 26

#### Exercise 8

Assuming that \lim_x\to4f(x) exists, find this limit if

\lim_x\to4(6x-2f(x)) = 26

#### Exercise 9

Assuming that \lim_x\to4f(x) exists, find this limit if

\lim_x\to4(2-3x + 10f(x)) = 0

#### Exercise 10

Assuming that \lim_x\to4f(x) exists, find this limit if

\lim_x\to4(2f(x) + x)-\lim_x\to4(f(x)-x) = 10