1.1 Limits

Math 135 Class Notes

1.1

Business Calculus

Spring 2009

Limits: A Numerical and Graphical Approach

The limit of a function is the fundamental concept in calculus and is used to define the derivative of a function, the subject of this first chapter. In this first section, we’ll introduce an intuitive definition of the limit of a function. Calculus is concerned with how function values, or outputs, change as the input changes. Suppose we have a function y = f (x). Then x is the input and y the output. Suppose that, as the input x gets closer and closer to some fixed number a, the function values get closer and closer to some fixed number L. The number L is called the limit of f as x approaches a. EXAMPLE

Consider the function defined by

x2 + x − 2 . x−1 First note that f (x) is not defined at x = 1 since the denominator x − 1 equals 0 at x = 1. Even though the function is not defined at x = 1, we can still examine how the function behaves for x close to 1. We can do this either numerically by constructing a table of function values for values of x close to 1, or graphically by looking at its graph near x = 1. a) Complete the following table of values f (x) =

x approaches 1 from left −→ | ←− x approaches 1 from right x

0

0.8

0.9

0.99

0.999

1

1.001

1.01

1.1

1.2

2

f (x) Based upon the table, what are the function values doing as x gets closer and closer to 1?

b) Use the table of values to sketch the graph of the function close to 1. Based upon the graph, what are the function values doing as x gets closer and closer to 1? y 4

3

2

1

–1

0

1

2

3

x

1

2

Chapter 1

Differentiation

In the above example, even though the function is not defined at x = 1, as x approaches 1 from either side, the function approaches 3. We use an arrow, →, to stand for the words “approaches.” So the above statement can be written: As x → 1, f (x) → 3. The number 3 is called the limit of f (x) as x approaches 1 from either side and we write lim f (x) = 3.

x→1

DEFINITION OF LIMIT As x approaches a, the limit of f (x) is L, written lim f (x) = L

x→a

provided that we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. The notation lim f (x) indicates that x approaches a from both the left and the right. If we only x→a allow x to approach a from the left or from the right, we can consider a one-sided limit. We write lim f (x) to indicate a limit from the right

x→a+

and lim f (x) to indicate a limit from the left

x→a−

EXAMPLE

Consider the piecewise-defined function given by Ω 2x + 2 for x < 1 H(x) = 2x − 4 for x ≥ 1

Complete the following table of values and draw a graph of the function to find the limit lim H(x), if x→1

it exists. If not, explain why the limit does not exist. x

0

0.8

0.9

0.99

0.999

f (x) y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1

1 –1 –2 –3 –4 –5 –6

2

3

4

5

6

x

1

1.001

1.01

1.1

1.2

2

1.1

3

Limits: A Numerical and Graphical Approach

In the preceding example, the function H(x) approaches different values as x approaches 1 from the left and from the right. The limits from the left and from the right both exist but are not equal to one another. In order for the (two-sided) limit lim f (x) to exist, both one-sided limits must exist and x→a be the same. THEOREM: RELATIONSHIP BETWEEN ONE-SIDED AND TWO-SIDED LIMITS The limit of f (x), as x approaches a, is L if and only if the limits from the left and the right both exist and both equal L. That is, lim f (x) = L x→a

if and only if lim f (x) = L

x→a−

and

lim f (x) = L.

x→a+

In the first example in this section, the limit lim f (x) exists even though the function f (x) is not x→1

defined at x = 1. In the second example, the left-hand limit lim− H(x) = 4 even though H(1) 6= 4. x→1

THE LIMIT DOES NOT DEPEND UPON THE FUNCTION VALUE The limit of a function as x approaches a only depends upon the function values close to a and not at a itself. This means that the limit at a does not depend upon f (a) or even on whether that function value exists. EXAMPLE

Consider the piecewise-defined function given by Ω 5 for x = 2 G(x) = x − 1 for x 6= 2

Graph the function and find each of the following limits, if they exist. If necessary state why the limit does not exist. y a) lim G(x) x→2

6 5 4 3 2 1

b) lim G(x) x→−3

–6 –5 –4 –3 –2 –1

1 –1 –2 –3 –4 –5 –6

2

3

4

5

6

x

4

Chapter 1

EXAMPLE

Differentiation

Consider the function given by

1 +3 x−2 The graph of the function is shown below. Complete the following table of values and use the table and graph to find the limit lim f (x), if it exists. If necessary, state why the limit does not exist. f (x) =

x→2

x

1

1.5

1.9

1.99

1.999

2

2.001

2.01

2.1

2.5

3

f (x) y 8 7 6 5 4 3 2 1 –4

–3

–2

–1

1

2

3

4

5

6

7

8

x

–1 –2 –3 –4

The one-sided limits in the preceding example are −∞ and ∞. Keep in mind that ∞ and −∞ are not real numbers. We use ∞ as a special notation to indicate a quantity which is increasing without bound in a positive direction and −∞ to indicate a quantity which is increasing without bound in a negative direction. In addition to having infinite limits as in the preceding example, we can also consider limits at infinity, denoted by lim f (x) or lim f (x). x→∞

EXAMPLE

Consider the function f (x) =

lim f (x).

x→−∞

1 + 3 in the preceding example. Find lim f (x) and x→∞ x−2

x→−∞

x f (x)

10

100

1000

10,000

x f (x)

−10

−100

−1000

−10,000