§2 Limits and continuity2.1 Some important definitions and results

about functions

Definition: Let be any set. A function is called a real-valued function on .

Definition: A function is called a function of a real variable, if and.

A function is a real valued function of a real variable, where .

• Note:

Definition: A function is said to be bounded above, if there exists for all where are subsets of .

Definition: A function is said to be bounded bellow, if there exists for all where are subsets of .

• Note:

Theorem 2.1.1: A function is bounded if and only if there exists such that for all where are subsets of .i.e. is bounded , Proof:

Definition: A function is said to be bounded, if it is both bounded above and bounded bellowwhere are subsets of .

2.2 Limit of a function

Given a function f(x), if x approaching 3 causes the function to take values approaching (or equaling) some particular number, such as 10, then we will call 10 the limit of the function and write

In practice, the two simplest ways we can approach 3 are from the left or from the right.

For example, the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3– , and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3+. Such limits are called one-sided limits.

Example:

We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.

2.3 One-Sided LimitWe have introduced the idea of one-sided limits. We write

lxfcx

)(lim

and call l the limit from the left (or left-hand limit) if is close to l whenever x is close to c, but to the left of c on the real number line.

and call m the limit from the right (or right-hand limit) if is close to m whenever x is close to c, but to the right of c on the real number line.

mxfcx

)(lim

We write

2.4 Methods of finding One-Sided Limit

2.4.1 Left-hand Limit:

lim𝑥→𝑐−

𝑓 (𝑥 )=limh → 0h>0

𝑓 (𝑐−h)

cc-h

h

Example:

2.4.2 Right-hand Limit:

c

h

𝑐+h

lim𝑥→𝑐+¿ 𝑓 (𝑥 )=lim

h→ 0h>0

𝑓 (𝑐+h )¿¿

Example:

2.5 The precise ( ) definition of Limit

We say that the limit of ( ) as approaches is and writef x x a L

lim ( ) x a

f x L

if the values of ( ) approach as approaches . f x L x a

a

L( )y f x

-

1: L be a real valued function of real variable. If given any for all , then for all

i.e. ,

Proof:

2: L be a real valued function of real variable. If given any for all , then for all , where

i.e. ,

Proof:

Some important results

The Definition of Limit-

lim ( ) We say if and only if x a

f x L

given a positive number , there exists a positive such that

if 0 | | , then | ( ) | . x a f x L

( )y f x a

LL

L

a a

such that for all in ( , ), x a a a

then we can find a (small) interval ( , )a a

( ) is in ( , ).f x L L

This means that if we are given a

small interval ( , ) centered at , L L L

Examples

21. Show that lim(3 4) 10.

xx

Let 0 be given. We need to find a 0 such that if | - 2 | ,x then | (3 4) 10 | .x

But | (3 4) 10 | | 3 6 | 3 | 2 |x x x

if | 2 |3

x

So we choose .3

1

12. Show that lim 1.

x x

Let 0 be given. We need to find a 0 such that 1if | 1| , then | 1| .x x

1 11But | 1| | | | 1| . x

xx x x

What do we do with the

x?

Limit Theorems

If is any number, lim ( ) and lim ( ) , thenx a x a

c f x L g x M

a) lim ( ) ( )x a

f x g x L M

b) lim ( ) ( ) x a

f x g x L M

c) lim ( ) ( )x a

f x g x L M

( )d) lim , ( 0)( )x a

f x L Mg x M

e) lim ( )x a

c f x c L

f) lim ( ) n n

x af x L

g) lim x a

c c

h) lim x a

x a

i) lim n n

x ax a

j) lim ( ) , ( 0)

x af x L L

The right-hand limit of f (x), as x approaches a, equals L

written:

if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a.

lim ( )x a

f x L

a

L( )y f x

One-Sided Limits

The left-hand limit of f (x), as x approaches a, equals M

written:

if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a.

lim ( )x a

f x M

a

M

( )y f x

lim ( ) if and only if lim ( ) and lim ( ) .x a x a x a

f x L f x L f x L

Theorem