Lesson05 - Limits Involving Infinity Slides

Section 1.5 Limits Involving Infinity; Asymptotes Math S-1ab Calculus I and II

June 28, 2007

Announcements I

We will hold off on §1.1 problems until I hear back from tech support.

Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L

x→∞

means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large.

Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L

x→∞

means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large.

Definition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L

x→∞

or

lim f (x) = L.

x→−∞

Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L

x→∞

means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large.

Definition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L

x→∞

y = L is a horizontal line!

or

lim f (x) = L.

x→−∞

Theorem Let n be a positive integer. Then 1 I lim =0 x→∞ x n 1 I lim =0 x→−∞ x n

Using the limit laws to compute limits at ∞

Example Find

2x 3 + 3x + 1 x→∞ 4x 3 + 5x 2 + 7 lim

if it exists. A does not exist B 1/2 C 0 D ∞

Using the limit laws to compute limits at ∞

Example Find

2x 3 + 3x + 1 x→∞ 4x 3 + 5x 2 + 7 lim

if it exists. A does not exist B 1/2 C 0 D ∞

Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 4x 3 + 5x 2 + 7 x 3 (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2

Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 4x 3 + 5x 2 + 7 x 3 (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2

Upshot When finding limits of algebraic expressions at infinitely, look at the highest degree terms.

Another Example

Example Find

√ lim

x→∞

3x 4 + 7 x2 + 3

Another Example

Example Find

√ lim

x→∞

Solution The limit is

√

3.

3x 4 + 7 x2 + 3

Example x2 . x→∞ 2x

Make a conjecture about lim

Example x2 . x→∞ 2x

Make a conjecture about lim

Solution The limit is zero. exponential growth is infinitely faster than geometric growth

Infinite Limits Definition The notation lim f (x) = ∞

x→a

means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a.

Definition The notation lim f (x) = −∞

x→a

means that the values of f (x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. Of course we have definitions for left- and right-hand infinite limits.

Vertical Asymptotes

Definition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following is true: I lim f (x) = ∞ I lim f (x) = −∞ x→a

I I

x→a

lim f (x) = ∞

I

lim f (x) = ∞

I

x→a+

x→a−

lim f (x) = −∞

x→a+

lim f (x) = −∞

x→a−

Infinite Limits we Know

1 =∞ x→0+ x 1 lim = −∞ x→0− x 1 lim 2 = ∞ x→0 x lim

Finding limits at trouble spots

Example Let f (t) =

t2 + 2 t 2 − 3t + 2

Find lim f (t) and lim+ f (t) for each a at which f is not t→a−

continuous.

t→a

Finding limits at trouble spots

Example Let f (t) =

t2 + 2 t 2 − 3t + 2

Find lim f (t) and lim+ f (t) for each a at which f is not t→a−

t→a

continuous.

Solution The denominator factors as (t − 1)(t − 2). We can record the signs of the factors on the number line.

−

0 1

+ (t − 1)

− −

+ (t − 1)

0 1 0 2

+ (t − 2)

−

+ (t − 1)

0 1

−

0 2 +

+ (t − 2) (t 2 + 2)

−

+ (t − 1)

0 1

−

0 2 +

1

+ (t − 2) (t 2 + 2)

2

f (t)

−

+ (t − 1)

0 1

−

0 2 +

(t 2 + 2)

+ 1

+ (t − 2)

2

f (t)

−

+ (t − 1)

0 1

−

0 2 +

+

±∞ 1

+ (t − 2) (t 2 + 2)

2

f (t)

−

+ (t − 1)

0 1

−

0 2 +

+

±∞ − 1

+ (t − 2) (t 2 + 2)

2

f (t)

−

+ (t − 1)

0 1

−

0 2 +

+

±∞ − ∓∞ 1 2

+ (t − 2) (t 2 + 2) f (t)

−

+ (t − 1)

0 1

−

0 2 +

+

±∞ − ∓∞ 1 2

+ (t − 2) (t 2 + 2) + f (t)

Limit Laws with infinite limits

I

The sum of positive infinite limits is ∞. That is ∞+∞=∞

I

The sum of negative infinite limits is −∞. −∞ − ∞ = −∞

I

The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞

Rules of Thumb with infinite limits

I

The sum of positive infinite limits is ∞. That is ∞+∞=∞

I

The sum of negative infinite limits is −∞. −∞ − ∞ = −∞

I

The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞

Rules of Thumb with infinite limits I

I

The product of a finite limit and an the finite limit is not 0. ( ∞ a·∞= −∞ ( −∞ a · (−∞) = ∞

infinite limit is infinite if if a > 0 if a < 0. if a > 0 if a < 0.

The product of two infinite limits is infinite. ∞·∞=∞ ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞

I

The quotient of a finite limit by an infinite limit is zero: a = 0. ∞

Indeterminate Limits

I

Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely.

Indeterminate Limits

I

I

Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely. 1 Limits of the form are also indeterminate. 0

Rationalizing to get a limit

Example Compute lim

x→∞

p  4x 2 + 17 − 2x .

Rationalizing to get a limit

Example Compute lim

x→∞

p  4x 2 + 17 − 2x .

Solution This limit is of the form ∞ − ∞, which we cannot use. So we rationalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on.