• What is “calculus”?

• What do you learn in a calculus class?

• How do algebra and calculus differ?

You will be able to answer all of these questions after you finish the course.

10.1 Introduction to Limits

• One of the basic concepts to the study of calculus is the concept of limit.

• This concept will help to describe the behavior of f(x) when x is approaching a particular value c.

• In this section, we will review and learn more about functions, graphs, and limits

Example 1a: f(x) = 2x - 1 Discuss the behavior of of f(x) when x is close to 2

using graph

• Graph f(x) = 2x - 1• When x is closer and

closer to 2,

F(x) is closer to 3

Therefore:

The limit of f(x) as x approaches 2 is 3

lim(2x-1) = 3 = f(2)X2

Example 1b: f(x) = 2x - 1 Discuss the behavior of the values of f(x) when x is close to 2

using table

• Find: lim (x+2) and lim (3x+1)X0 X -1

Do you get 2 and -2? If not, try again

Example 2: Discuss the behavior of f(x) when x is closer to 2

2

4)(

2

x

xxf

x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5

f (x) 3.5 3.9 3.99 3.999 ? 4.001 4.01 4.1 4.5

If x = 2, f(x) is undefined. If you graph, you will see a hole there.

Therefore, when x is closer and closer to 2, f(x) is closer to 4

lim f(x) = 4 = f(2) or

X2

Example 2: Discuss the behavior of the values of f(x) when x is closer to 2.

Does the limit exist?

2

2)(

x

xxf

x 0 1 1.9 1.99 2 2.001 2.01 2.1 2.5

f (x) -1 -1 -1 -1 ? 1 1 1 1

* This function is not defined when x = 2.

* The limit does not exist because the limit on the left and the limit on the right are not the same.

Lim f(x) = -1 represents the limit on the left of 2

Lim f(x) = 1 represents the limit on the right of 2

X2 -

X2 +

■ We write

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

■ We write

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

■ In order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

Example 3

(A) Discuss the behavior of f(x) for x near 0

Lim f(x) =

Lim f(x) =

Lim f(x) =

F(0) =

X0 -

X 0 +

X0

0

0

0

0

(B) Discuss the behavior of f(x) for x near 1

Lim f(x) =

Lim f(x) =

Lim f(x) =

F(1) =

X1 -

X 1 +

X1

Example 3 - continue

1

2

does not exist

not defined

(C) Discuss the behavior of f(x) for x near 3

Lim f(x) =

Lim f(x) =

Lim f(x) =

F(3) =

X3 -

X 3 +

X3

Example 3 - continue

3

3

3

not defined

Lim (x2 + 5x)

= lim x2 + lim 5x

= (lim x) (lim x) + 5 (lim x)

= (-2) (-2) + 5 (-2)

= 4 – 10 = -6

X-2

Example 4

X-2 X-2

X-2 X-2 X-2

Property 1

Property 4 and 3

Try lim (x4 + 3x – 2)X-1 If you don’t get -4, try again

Example 5

Lim

=

=

63 2 xX2

X2

Property 8

Try limX-1

)63lim( 2 x

66126)2)(2(3

22 2 x If you don’t get 2, try again

Example 6

1lim

2

2

2 x

xx

)1(lim

lim2

2

2

2

x

x

x

x

5

4

1)2)(2(

)2)(2(

Note that this is a rational function with a nonzero denominator at x = -2

= =

2lim

41 x

xTry

x If you don’t get 1/3, try again

Example 7

12

32)(

x

xxf

If x < 5

If x > 5

A)Lim f(x) =

lim (2x+3) = 2(5)+ 3 = 13X5-

X5-

B) Lim f(x) =

lim (-x+12)=-5+12=7 X5+

X5+

C) Lim f(x) =

Does not exist because the left hand the right hand limits are not equalX5

D) F(5) =

is not defined

Example 8: Use algebraic and/or graphical techniques to analyze each of the following

indeterminate forms

2

2

1

2

2

1

21

)1(

1lim

1

)1(lim

1

1lim

x

x

x

x

x

x

x

x

xA)

C)

B)

See next page for step by step instruction

Example 8 - Solutions

Note: when you find the limits of the above problems, you must factor first and then simplify before you substitute the number for x

Different Quotient (pre-cal)

h

xfhxf )()(

Indeterminate form

Examples:

• Find the following limit for the following functions

h

fhfxxf

h

fhfxxf

h

fhfxxf

h

h

h

)3()3(lim;)(

)1()1(lim;1)(

)4()4(lim;27)(

0

0

0

9)

10)

11)

See the next pages for step by step instruction

Because the limit on the left and the litmit on the right are not the same. Therefore, this limit does not exist

Rationalize the numerator