Section 1.8Limits

Consider the graph of f(θ) = sin(θ)/θ

• Let’s fill in the following table

• We can say that the limit of f(θ) approaches 1 as θ approaches 0 from the right

• We write this as

• We can construct a similar table to show what happens as θ approaches 0 from the left

θ 0.5 0.4 0.3 0.2 0.1 0.05

sin(θ)/θ

1sin

lim0

θ 0.5 0.4 0.3 0.2 0.1 0.05

sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995

θ -0.5 -0.4 -0.3 -0.2 -0.1 -0.05

sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995

• So we get

• Now since we have we say that the limit exists and we write

1sin

lim0

1sin

limsin

lim00

1sin

lim0

A function f is defined on an interval around c, except perhaps at the point x=c. We define the limit of f(x) as x approaches c to be a number L, (if one exists) such that f(x) is as close to L as we want whenever x is sufficiently close to c (but x≠c). If L exists, we write Lxf

cx

)(lim

Note: If f(x) is continuous at c, than

so the limit is just the value of the function at x = c

lim ( ) ( )x c

f x f c L

We define _______to be the number L, (if one

exists) such that for every ε > 0 (as small as we

want), there is a δ > 0 (sufficiently small) such

that if |x – c| < δ and x ≠ c, then |f(x) – L| < ε.

)(lim xfcx

L + ε

L - ε

L

c-δ c c+δ

ε

ε

f(x)

Properties of Limits

)(lim)(lim))()((lim xgxfxgxfcxcxcx

))(lim())((lim xfbxbfcxcx

))(lim))((lim())()((lim xgxfxgxfcxcxcx

,)(lim

)(lim)(lim )(

)(

xg

xf

cx

cxxgxf

cx

kkcx

lim cxcx

lim

Compute the following limits

2

1lim

2sinlim

1

3lim

1

1lim

21

3

2

xx

xx

xx

xx

x

x

x

x

• Let’s take a look at the last one

• What happened when we plugged in 1 for x?

• When we get we have what’s called an

indeterminate form

• Let’s see how we can solve it

2

1lim

21

xx

xx

0

0

• Let’s look at the graph of

• Seems to be continuous at x = 1

2

1lim

21

xx

xx

When does a limit not exist?

)(lim)(lim00

xfxfxx

When

Example

2for

2for)(

2 xx

xxxf

Limits at Infinity• If f(x) gets sufficiently close to a number L

when x gets sufficiently large, then we write

• Similarly, if f(x) approaches L when x is negative and has a sufficiently large absolute value, then we write

• The line y = L is called a horizontal asymptote

Lxfx

)(lim

Lxfx

)(lim

Limits at Infinity• Let’s show the following function has a limit,

and thus a horizontal asymptote.

• So we need to calculate

3

2 3

5 2 1lim

2 3 7x

x x

x x x

3

2 3

5 2 1( )

2 3 7

x xf x

x x x

Examples

2

122lim

1723

53lim

2

2

x

x

x

x

e

e

xx

xx

Formal Definition of Continuity

The function f is continuous at x = c if f is defined at x = c and if

)()(lim cfxfcx