Understanding Limits in Calculus, Lecture notes of Statistics

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Unit 2: Limits

OUTCOMES

At the end of this unit you should be able to:

 Explain what is meant by a limit.

 Find the value of a limit.

 Give examples of one-sided limits.

Unit 2: Limits

Definition of a limit

Let us discuss the following before defining a limit.

Let us first get a “feeling” for limits by discussing an example.

Consider the function when is near to two but not equal to two.

If we tabulate values of which approach 2 with the corresponding values

of the function we observe that the closer comes to the value 2 the

closer comes to the value 3.

f ( ) x  2 x  1

x

f ( ) x

x f ( ) x

Unit 2: Limits

Techniques for finding limits

Rule 1 and 2 for Limits:

, where and are real numbers

, where is a real number

lim

x a

C C





a

C

lim

x a

x a





a

Unit 2: Limits

Example: Evaluate the following

i.

ii.

Solution

i. ii.

lim 5

x  a

1

lim 2

x

x



lim 5 5

x  a



1

lim 2 2(1) 2

x

x



 

Unit 2: Limits

Example

1.

2.

0 0 0

lim 1 lim lim1 (0) 1 1

x x x

x x

  

2 2 2

1 1 1

lim 2 lim lim 2 (1 ) 2 1

x x x

x x

  

Unit 2: Limits

Rule 4: Limit of a Product

 

lim ( ). ( ) lim ( ).lim ( )

x a x a x a

f x g x f x g x

  

1 2

 L L.

This rule states that the limit of the product of two functions is the product of

their limits.

Unit 1: Function Notation

Rule 5: Limit of a Quotient

if

1

2

lim ( )

( )

lim ,

( ) lim ( )

x a

x a

x a

f x

f x L

g x g x L







 

2

L  0

If the limit of the denominator is not zero, then the limit of the quotient of

two functions is the quotient of their limits.

Unit 1: Function Notation

Example

 

   

2 2 2

2

3

lim (4 2 ) 4 2 4 2(3 )

2 1 lim (2 1) 2(3) 1 7

lim

x

x

x x

x x

x





  

Unit 1: Function Notation

Example

1.

2.

   

 

5

5 5

2 2

lim lim ( ) 2 32

x x

x x

 

    

3 3

3

2 2

lim 4 lim(4 ) 8 2

x x

x x

 

  

Unit 1: Function Notation

Activity 2

Evaluate the following