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Worksheet Name Topic: Limits _________

Introduction to limits

In this section, we will learn about limits and how to find the limits of a function either at a

certain point or at infinity.

Consider the two graphs of the function 1)(2 x x x f   and 1 ,

1

1)(

3

x

 x

 x xg below.

Now let’s investigate the similarities and the differences between the graphs of these two

functions.

SIMILARITIES DIFFERENCES

The two graphs both have similar shape (like a

parabola)1.  1)(

2 x x x f    is defined when

x = 1, and the value is 3.

2.  1 ,1

1)(

3

x

 x

 x xg  is not defined 

when x = 1.

Although we know that g( x) is not defined when  x = 1, we can use a mathematical technique

called “limit(s)” to guess or estimate the value of the function g( x) when  x = 1. This technique

called limit can help us to investigate the behavior of the function g( x) by considering values of  x that are close to 1 but not equal to 1. Through this investigation we can be able to make a guess

or come out with an estimate of the value of the function g( x) when x = 1. In general, the word

limit is a mathematical technique for  predicting (or guessing) the value of a function as itsindependent variable approaches some fixed number or infinity.

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To find the limit of a function at a point by way of table of values, start your investigation fromone point below the point in question and one point above the point in question. For example, to

find 3lim x f(x), the point in question is 3, therefore start the investigation from 2 and choose valuesof  x from the interval [2, 3), and then start another investigation from 4 by choosing values of  x 

from the interval (3, 4]. [2, 3) means 2 is included  in the interval but 3 is excluded  from the

interval and (3, 4] means 3 is excluded from the interval but 4 is included in the interval.

Example

Find1

1lim

2

1

 x

 x

 xby means of a table of values. Note that x cannot be 1.

Alternatively, you can setup a table like the one below and determine the limits from both sides. x 0 0.5 0.9 0.99 0.999 1 1.001 1.01 1.1 1.5 2

1

12

 x

 x 

1 1.5 1.9 1.99 1.999 1

1lim

2

1

 x

 x

 x 

= 2

2.001 2.01 2.1 2.5 3

 The Numerical approach (Table of values)

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EXAMPLE: Lets consider the function f(x)  = x2 

Height Limit

Different values of x have different height.

The height changes throughout the

functions domain.

As x gets closer and closer to 2, but not

equal to 2, what height does f(x) gets

closer and closer to?

To find the height at x = 2, we simply put

x = 2 in the function. The function reaches

a height of 4 at x = 2.

We notice that whenever we are getting

closer and closer to the x value of 2, the

function is getting closer and closer to theheight of 4.

Height: f (2) = 4 Limit:2

lim x

f(x) = 4.

 The Graphical approach

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EXAMPLE: Lets consider the function2

86)(

2

 x

 x x xg  

Height Limit

The graph has a “hole” at x = 2. The “hole”

exists because if we put x = 2 into thefunction, the result will be division by zero,

which is mathematically impossible.

As x gets closer and closer to 2, but not equal to

2, what height does g(x) gets closer and closerto?

To find the height at x = 2, we simply put x

= 2 in the function. The function does not

have a value at x = 2.

Although there is a hole, the function still

possesses a limit. We notice that whenever we

are getting closer and closer to the  x value of 2,the function is getting closer and closer to the

height  of -2. Clearly, this function intends to

reach the height of 2, even though the height of 

2 does not exist.

Height: g (2) = does not exist Limit:2

lim x

g(x) = -2.

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EXAMPLE: Lets consider the function

2,1

2,2

86

)(

2

 x

 x x

 x x

 xg  

Height Limit

The graph has a “hole” at x = 2, however,

the function has a height of -1 at x = 2.

As x gets closer and closer to 2, but not equal to 2,

what height does g(x) gets closer and closer to?

We notice that whenever we are getting closer and

closer to the  x value of 2, the function is getting

closer and closer to the height of -2. Clearly, this

function intends to reach the height of 2, even

though the actual height of the function at x = 2 is-1.

Height: g (2) = -1 Limit:2

lim x

g(x) = -2.

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EXAMPLE: Lets consider the function whose graph is represented below

Height Limit

The graph has two “holes” at x = 2. As x gets closer and closer to 2, but not equal

to 2, what height  does g(x) gets closer and

closer to?

We notice that whenever we are getting closerand closer to the  x value of 2 from the left  

side, we are convinced that the function isgetting closer and closer to the height  of 1.

However, whenever we are getting closer and

closer to the  x value of 2 from the right side,we are convinced that the function is getting

closer and closer to the height  of 2. Clearly,

this function intends to reach two different

heights, which is impossible for any function.

Height: f (2) = does not exist Limit:2

lim x

f(x) = does not exist

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Let’s consider the function whose graph is repr esented below.

If we inspect the graph, we can make the following observations:

1)  We can see that there is a break at x = 4.

2)  If we travel towards x = 4 from the right, we will arrive at the height of 2.

The height at which we arrive at if we are traveling from the right is called the Right-Hand

Limit of f( x) as x approaches a specific value of  x, in this case as x approaches 4. We denote

a Right-Hand Limit with a little positive symbol (+) at the location where the exponent of 

the number should be. This is shown on the graph below.

Left-Hand Limits and Right-Hand Limits 

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3)  If we travel towards x = 4 from the left, we will arrive at the height of 1.

The height at which we arrive at if we are traveling from the right is called the Left-Hand

Limit of f( x) as x approaches a specific value of  x, in this case as x approaches 4. We denote

a Left-Hand Limit  with a little negative symbol (-) at the location where the exponent of thenumber should be. This is shown on the graph below.

4) We see that the heights do not march. This means that there is no limit for the function f( x) at x = 4. For a limit to exist, both the right and left hand limits must be equal.

In general,

c xlim f (x) exists if the following all of the following statements are true;

1.

c xlim f (x) is definite and has a value.

2.

c xlim f (x) is definite and has a value.

3.

c xlim f (x) =

c xlim f (x).

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Given the graph of  f(x) shown below,

determine the limit a) at x=0  and b) at x=1. 

Solution

a) at x=0

Left-hand Limit Right-hand Limit  Two-sided Limit Value of Function

0

lim x

 f (x) = 0

0

lim x

 f (x) = 0

From Left-hand and Right

hand we conclude that:

0lim x

 f (x) = 0

At x= 0,

 f (0) = 1

b) at x= 1

Left-hand Limit Right-hand Limit  Two-sided Limit Value of Function

1

lim x

 f (x) = 1

1

lim x

 f (x) = 0

From Left-hand and Right

hand we conclude that:

1lim x

 f (x) = Does Not

Exist 

At x= 1,

 f (1) = 1

Example

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Given the function f(x) =

,32

,3

 x

 x

1

1

 x

 xdetermine whether or not the limit exist at x=1.

Solution

The domain of this function is all real numbers. This is just one function, but it has different

description in different parts of the domain.

Left-hand Limit Right-hand Limit  Two-sided Limit Value of Function

1lim x

 f (x) =

1lim x

3 x

= 3

1lim x

 f (x) =

1lim x

2 x+3

= 5

From Left-hand and Right

hand we conclude that:

0lim x

 f (x) = Does Not

Exist at x = 0 

At x= 1,

 f (1) = 3(1) = 3

Given the function f(x) =

,12

,3

 x

 x

1

1

 x

 xdetermine whether or not the limit exist at x=1.

Solution

The domain of this function is all real numbers. This is just one function, but it has different

description in different parts of the domain.

Left-hand Limit Right-hand Limit  Two-sided Limit Value of Function

1

lim x

 f (x) =

1

lim x

3 x

= 3

1

lim x

 f (x) =

1

lim x

2 x+1

= 3

From Left-hand and Righthand we conclude that:

1lim x

 f (x) = 3

The limit exist at x = 1.

At x= 1,

 f (1) = 3(1) = 3

 An example

 Another example of limits

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Let f(x) =

2x ,7

2x,2

1032

 x

 x x

 

Determine whether or not the limit exist at x=2.

Solution

The domain of this function is all real numbers.

Left-hand Limit Right-hand Limit  Two-sided Limit Value of 

Function

2

lim x

 f (x) =

2

lim x 2

1032

 x

 x x 

=2

lim x )2(

)5)(2(

 x

 x x 

=2

lim x

( x + 5) = 7

Same as Left-handlimit:

2

lim x

 f (x) = 7

From Left-hand andRight hand we

conclude that:

2lim x  f (x) = 7

The limit exist atx = 2

At x= 2,

 f (2) = 7

 Another Example

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Determine whether or not the function f(x) = | x – 2 | has a limit at x = 2.

Solution:

The given function is the same as f(x) =

),2(

,2

 x

 x

2

2

 x

 x 

The domain of this function is x (all real numbers). This is just one function, but has different

description in different parts of the domain.

Left-hand Limit Right-hand Limit  Two-sided Limit Value of Function

2

lim x

 f (x) =

2

lim x

-(x-2) 

= 0

2

lim x

 f (x) =

2

lim x

(x-2)

= 0

From Left-hand and Right

hand we conclude that:

2lim x

 f (x) = 0

The limit exist at x = 0.

At x= 2,

 f (2) = (2-2) = 0

 Another example

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Limits of functions can be evaluated by using many different methods including table of values,

graphs, direct substitution, factorization, and rationalization. In this section, we will discuss the

above mentioned techniques.

To find the limit of a function, first try direct substitution. We usually use direct

substitution when the function you are dealing with is a polynomial, a rational, or any

algebraic function whose domain do not exclude the specific value that x approaches.

Direct substitution is always valid for polynomials and rational functions with nonzero

denominators. Do not use direct substitution if it makes a denominator zero.  

If p is a polynomial function, then

c xlim p(x) = p(c).

If r  is a rational function given by r (x) =)(

)(

 xg

 x f and g(c) 0, then

c xlim r (x) =

)(

)(

cg

c f  

Example

Find the limit: a)1

lim x

3 x4  – 2 x

2+ x  – 1 b)

2lim x 2

22

 x

 x x 

Solution:  Use direct substitution 

a) 1lim x 3 x

4

  – 2 x2

+ x  – 1 = 3(1)4

  – 2(1)2

+ (1) – 1 = 1

b)2

lim x 2

22

 x

 x x= 1

4

4

22

2222

 

Direct Substitution Technique

 Techniques for finding the value of a limit

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 If the limit of f(x) as x approaches c cannot be evaluated through direct substitution, because this

will cause denominator to be zero, then try to factor and cancel out the common factors  , and 

then use direct substitution again.

Example

Find the limit: a)1

lim x 1

12

 x

 xb)

3lim

 x 3

62

 x

 x x 

Solution: Since this is a rational function and the limit of the denominator is 0, try to

factor both numerator and denominator and cancel out the common factors.

a)1

lim x 1

12

 x

 x=

1lim x

211)1(lim)1(

)1)(1(

1

 x x

 x x

 x 

b)3

lim x 3

62

 x

 x x=

3lim

 x523)2(lim

)3(

)2)(3(

3

 x x

 x x

 x 

  If the numerator approaches 0 and the denominator also approaches 0 and the numerator 

involves a square root expression, then   rationalize the numerator by multiplying both the

 numerator and the denominator by the conjugate of the numerator. In general if f(x) = g(x) for every x except possibly x = c, in some interval containing c, then

c xlim f(x) =

c xlim g(x) = g(c)

Rationalization Technique

Factorization and cancelling the denominator Technique

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Example

Find the limit:4

lim x 4

2

 x

 x 

Solution:

f(x) =4

2

 x

 x 

Multiply numerator and denominator by conjugate  2 x .

g(x) =

2

1

)2)(4(

4

)2)(4(

422

2

2

4

22

 x x x

 x

 x x

 x x x

 x

 x

 x

 x 

Thus,c x

lim f(x) =c x

lim g(x) implies

4lim x 4

2

 x

 x=

4lim x 4

1

22

1

24

1

2

1

 x 

Use the graph below to answer the following questions.

 Another example

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i h l A Introduction to Limits P 19

c x

lim k = k 

The limit of a constant is a constant.

Example

Find the limit: 4

lim x

12.

Solution:

4lim x

12 = 12.

 The limit of a constant