limits and their properties

Limits and Their Properties

Calculus Chapter 1

An Introduction to Limits

Calculus 1.1

Calculus Chapter 1 3

Calculus is…• The mathematics of change

• Velocity• Acceleration

• The mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, etc., that enable scientists, engineers, and economists to model real-life situations.


Calculus is …• A limit machine with three stages

1. Precalculus2. Limit process3. Calculus formulation

• Derivatives• Integrals


Tangent Line Problem• Except for vertical tangent lines, to find

the tangent line you must simply find its slope• You already know a point


Secant line• Used to approximate slope of tangent

line• A line through the point of tangency (P)

and a second point on the curve (Q)

, ( )P c f c

, ( )Q c x f c x


Slope of secant line

• As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line.

• slope of the tangent line is said to be the limit of the slope of the secant line

sec

f c x f c f c x f cm

c x c x


Limit• If f(x) becomes arbitrarily close to a

single number L as x approaches c from either side, the limit of f(x) as x approaches c is L

limx cf x L


Example

• What happens at x = 2?• To get an idea, look at values close to 2 from the left

and right

22

2lim4x

xx

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x)


Important to remember• The existence or nonexistence of f(x) at

x = c has no bearing on the existence of the limit of f(x) as x approaches c.


Example

3

3lim

0 3x

x xf x

x

x 2.9 2.99 2.999 3.001 3.01 3.1

f(x)


Limits that fail to exist• Behavior that differs from the right and

the left• Unbounded behavior• Oscillating behavior

• There are others


Example

0limx

xx

• If x is positive, f(x) = 1• If x is negative, f(x) = -1• No matter how close we get to 0, there will always be

negative 1 on the left and positive 1 on the right


Example20

1limx x

• As x gets closer to zero from either side, f(x) gets larger and larger

• “increases without bound”• Limit does not exist


Example

0

1limsinx x

x 2/p 2/3p 2/5p 2/7p 2/9p 2/11p

f(x)


Example cont’d

• See page 65• You can’t always trust the picture your

calculator draws• It’s wrong, but you can probably still

tell there is not a limit


Note• When we write

• We imply that the limit exists and the limit is L.

• If the limit of a function exists, it is unique.

limx cf x L

Properties of Limits

Calculus 1.2


Direct substitution• Works for some functions

• Called continuous at c• When

• This section – all limits can be evaluated this way

limx cf x f c


Basic limits

limx cb b

limx cx c

lim n n

x cx c


You try4

lim 6x

12limx

x

4

2limxx


Properties of limits• Page 71• Can be used on all limits, even those

that can’t be evaluated by direct substitution


Examples 2

1lim 1x

x

3

2lim2x x


Limits with radicals• Let n be a positive integer. The

following is valid for all c if n is odd, and is valid for c > 0 if n is even.

lim n n

x cx c


Limit of a composite function• If f and g are functions such that

limx cg x L

lim

x Lf x f L

limx cf g x f L


Examples

• Find

lim 27x cf x

3limx c

f x

lim

18x c

f x


Limits of trig functions• All can be evaluated by direct

substitution• Page 74


Example

2

0lim cosx

x

Techniques for Evaluating Limits

Calculus 1.3


Indeterminate form• Direct substitution yields 0/0• Can’t find limit directly


Functions that agree at all but one point

• If function is undefined at point c, find another function that gives the same values for all other points, and is defined at point c.• Cancellation• Rationalization


Example - cancellation

22

2lim4x

xx


You try

2

1

2 3lim1x

x xx


Example - rationalization

0

2 2limx

xx


You try

3

1 2lim3x

xx


The Squeeze Theorem• Page 80


Example

0

sinlimx

xx


Example

0

1 coslimx

xx


Two special trig limits

0

sinlim 1x

xx

0

1 coslim 0x

xx


Limits with trig functions• Try to write them using one of the two

special trig forms.


Example

0

3 1 coslimx

xx


Example

0

cos tanlimx

x xx


Example

2

0

tanlimx

xx

Continuity and One-Sided Limits

Calculus 1.4


Continuity• A function is continuous at x = c if

• There is no interruption in the graph• The graph is unbroken• There are no holes, jumps or gaps


Continuity• A function is continuous at c if

1. is defined

2. lim exists

3. limx c

x c

f c

f x

f x f c


Continuity over an open interval• Continuous on open interval if

continuous at each point in the interval


Discontinuities• When f is not continuous at c, it has a

discontinuity at c• Removable discontinuity if

• f can be made continuous by defining or redefining f(c)

• Nonremovable


Discontinuities• See page 85• Check for discontinuities where a

function is undefined or in a piecewise function where the definition of the function changes


Limit from the right• x approaches c from values greater

than c

limx c

f x L


Limit from the left• x approaches c from values less than c

limx c

f x L


One-sided limits• Useful for taking limits of functions

involving radicals

0lim 0n

xx


One-sided limits• Useful for step functions• See page 86

greatest integer such that x n n x


One-sided limits• Not “official” limits• If the limit from the left is not equal to

the limit from the right, the limit does not exist – theorem 1.10

• Unless “from the right” or “from the left” is specified, it means from both directions


Continuity over a closed interval• Continuous over a closed interval [a, b]

if it is continuous over the open interval (a, b) and

limx a

f a

limx b

f b


Example• Discuss the continuity of

24f x x


Properties of Continuity• Page 89• Correspond to properties of limits


Functions that are continuous at every point in their domains

• Polynomial functions• Rational functions• Radical functions• Trigonometric functions


Continuity of a composite function

If is continuous at and is continuous at g ,

then the composite function

is continuous at .

g cf c

f g x f g x c


Testing for continuity• Describe the intervals on which each

function is continuous

tan2xf x

3f x x x


Intermediate Value Theorem• See page 91• Must be continuous on closed interval• There may be more than one possibility


Intermediate Value Theorem• Useful for finding the zeros of a function

• If function is continuous over an interval and the value of the function changes in sign over the interval, the graph must cross y = 0 somewhere in the interval


Example• Use the intermediate value theorem to

show that the following function has a zero on the interval [0, 1]

3 3 2f x x x

Infinite Limits

Calculus 1.5


Infinite limits• f(x) increases or decreases without

bound as x approaches c• Approaches a vertical asymptote• Limit fails to exist

lim

limx c

x c

f x

f x


Vertical asymptotes• Occur when the denominator is zero,

but the numerator is not• Theorem 1.14• Page 98


Properties of Infinite Limits• Page 100


Example• Find the vertical asymptotes (if any) of

the function

3

42

f xx


Example• Find the vertical asymptotes (if any) of

the function.

2

3 2

42 2xf x

x x x


Example• Determine whether the function has a

vertical asymptote or a removable discontinuity at x = –1

2 6 7

1x xf xx


Example• Find the limit

3

21

1lim1x

xx x

limits and their properties

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