LIMITSAn Introduction to Calculus

LIMITSLIMITS

• What is Calculus?

• What are Limits?

• Evaluating Limits – Graphically– Numerically– Analytically

• What is Continuity?

• Infinite Limits

This presentation

Bring your graphing calculator to class!

WHAT IS CALCULUS?WHAT IS CALCULUS?

• Calculus is – the mathematics of change– dynamic, whereas earlier mathematics is static– a limit machine

PrecalculusMathematics

LimitProcess

Calculus

As an example, an object traveling at a constant velocity can be modeled with precalculus mathematics. To model the velocity of an accelerating object, you need calculus.

TWO CLASSIC CALCULUS PROBLEMSTWO CLASSIC CALCULUS PROBLEMSthat illustrate how limits are used in calculus

• The Tangent Line ProblemFinding the slope of a straight line is a precalculus problem.

Finding the area of a rectangle is a precalculus problem.

The limit process: Compute the slope of a line through P and another point, Q, on the curve. As the distance between the two points decreases, the slope gets more accurate.

.Q

• The Area Problem

The limit process: Sum up the areas of multiple rectan-gular regions. As the number of rectangles increases (and their width decreases) the approximation improves.

Finding the slope of a curve at a point, P, is a calculus problem.

Finding the area under a curve is a calculus problem.

See other examples on pages 42-43 of your textbook.

Limits: An Informal Definition

SO… If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit of f(x) as x approaches c is L.

Lxfcx

)(limThe limit itself is a value of the function, i.e., its y value. In this case, the limit is L.

The limit is taken as the input, x, approaches a specific value from either side. In this case, x is approaching c.

Limits: An Informal Definition

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.

Lcf )(The existence or nonexistence of f(x) at c has no bearing on the existence of . In fact, even if , the limit can still exist.

)(lim xfcx

Lxfcx

)(lim

IMPORTANT POINTIMPORTANT POINT:

1

1

,4

,2)(

x

xxf

-5 5

4

-4

2)(lim1

xfx

4)1( feven though

1,2)( xxf

2)(lim1

xfx

-5 5

4

-4

even though f(1) is undefined

f(c) is undefined f(c) ≠ L

Limits: Formal Definition

If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit of f(x) as x approaches c is L.

Lxfcx

)(lim

The ε-δ definition of a limit: ε (epsilon) and δ (delta) are small positive numbers.

means that f(x) lies in the interval (L – ε, L + ε)i.e., |f(x) – L| < ε.

means that x lies in the interval (c – δ, c + δ)i.e., 0 < |x – c| < δ. ↑ means x ≠ c

FORMAL DEFINITION:Let f be a function on an open interval containing c (except possibly at c) and let L be a real number. The statement

means that for each ε > 0, there exists a δ > 0 such that if 0 < |x – c| < δ, then |f(x) – L| < ε.

This formal definition is not used to evaluate limits.Instead, it is used to prove other theorems.

Lxfcx

)(lim

We will evaluate limits

• Graphically

• Numerically

• Analytically

Evaluating Limits Graphically

22)( 2 xxxf1.

(2, 6)

2,22)( 2 xxxxf2.

6)(lim2

xfx

Once again, the existence of a limit at a given point is independent of whether the function is defined at that point.

What is ?)(lim2

xfx What is ?)(lim

2xf

x

a parabola a parabola w/ a hole in it

(2, 6)

6)(lim2

xfx

Evaluating Limits Graphically (cont)

.)1(

)1)(3()(

x

xxxf

2)(lim1

xfx

)1(

34)(

2

x

xxxf3.

-5 5

4

-4

(-1, 2)

This rational function is undefined at x = -1, because when x is -1, the denominator goes to zero.

If you factor the numerator, you get

Canceling like factors, you get .3)( xxf

However, x = -1 is not a vertical asymptote*

This is a straight line. f(x) is still undefined at x = -1,so there is a hole in the graph at the point (-1, 2).

Even though f(-1) is undefined, is still 2. )(lim1

xfx

Rational functions can have point discontinuities as well as vertical asymptotes. If the rational expression can be simplified, the function is still not defined at this value of x, but the discontinuity becomes a point rather than a vertical line.

*

Graph a function with a given limit

Sketch any function that approaches (3, -1) from both directions. Simplest example is a straight line, e.g. f(x) = -x + 4.

-5 5

4

-4

(3, -1)

f(3) = -1

-5 5

4

-4

(3, -1)

f(3) is undefined

-5 5

4

-4

(3, 3)

1)3( f

Three cases:

Draw the graph of a function such that .1)(lim3

xfx

Limits that do not exist

Limits do not exist when:

1. f(x) approaches a different number from the right side of c than it approaches from the left side.

2. f(x) increases or decreases w/o bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

We will consider these cases graphically.

Evaluating Limits Graphically Limits that do not exist

dnexfx

)(lim0

x

xxf

||)( 5.

This function is undefined at x = 0, because the denominator goes to zero.

-2.5 2.5

2

-2

When x > 0, . 1||

x

x

Approaching zero from the right, . 1)(lim0

xfx

When x < 0, . 1||

x

x

Approaching from the left, . 1)(lim0

xfx

When the behavior differs from right and left, the limit does not exist.

f(x) approaches a different number from the right side of c than it approaches from the left side.

(0, 1)

This is called the right-hand limit.

(0, -1)

This is called the left-hand limit.

Evaluating Limits Graphically Limits that do not exist

dnexfx

)(lim3

3

2)(

xxf6.

This function is undefined at x = 3, because the denominator goes to zero. It can not be simplified, so there is a vertical asymptote at x = 3.

Approaching 3 from the right, f(x) increases without bound.

When the function increases or decreases without bound, the limit does not exist.

f(x) increases or decreases without bound as x approaches c.

Approaching 3 from the left, f(x) decreases without bound.

Lxfcx

)(lim

We will evaluate limits

• Graphically

• Numerically

• Analytically

Evaluating Limits Numerically

)1(

1lim

2

1

x

x

x

We will do this using the Ti89 and its table function.

Ex 1: Find

Set . Graph the function. What happens to f(x) as x

approaches 1 from the left and from the right?)1(

12

x

xy

Use the TBLSET function. Set TblStart = 0.8 and set Tbl = 0.01. Go to the table.

What happens as you move down the table from x = 0.8 to x = 1?

Now set TblStart = 1.2.What happens as you move up the table from x = 1.2?

In both cases, the f(x) values are getting closer and closer to 2, therefore

21

1lim

2

1

x

x

x

Evaluating Limits Numerically23 )3(

1lim

xxEx 2: Find

Set . Graph the function. What happens to f(x) as x

approaches 3 from the left and from the right?

2)3(

1

xy

Use the TBLSET function. Set TblStart = 2.8 and set Tbl = .01. Go to the table.

What happens as you move down the table from x = 2.8 to x = 3?

Now set TblStart = 3.2.What happens as you move up the table from x = 3.2?

In both cases, f(x) values are growing without bound as x approaches 3 from the left and from the right. Therefore we say

23 )3(

1lim

xxdoes not exist

LIMITSLIMITS• Limits are a fundamental part of Calculus• The limit of a function at a specific input value, c, is the

value of the function as you get increasingly closer to c.– The function need not be defined at this input value, c.– If the function is defined at c, the limit value and function value

need not be the same. • Limits can be evaluated

– Graphically, – Numerically, and – Analytically

• Limits do not exist if,– The function approaches different values from the right and left– The function increases or decreases without bound– The function oscillates between 2 fixed values

• Today, we focused on evaluating limits by studying the graphs of functions and numerically approaching x from both directions.

• Next we will be evaluating limits analytically.

Homework

• Pg. 47 (1-8)

• Pg. 54 – 55 (6,8, 9-18)