limits at infinity (3.5)

limits at infinity (3.5)

December 20th, 2011December 20th, 2011

I. limits at infinityDef. of Limit at Infinity: Let L be a real number.1. The statement means that for each there exists an M>0 such that whenever x>M.

2. The statement means that for eachthere exists an N<0 such that whenever x<N.

limx→ ∞

f(x) =L ε > 0

f (x)−L < ε

limx→ −∞

f(x) =L ε > 0

f (x)−L < ε

II. horizontal asymptotes

Def. of a Horizontal Asymptote: The line y=L is a horizontal asymptote of the graph of f if or .

limx→ ∞

f(x) =L limx→ −∞

f(x) =L

*In other words, when a function f has a real number limit as or , it means that the function is approaching a horizontal asymptote at that limit value.

x→ ∞ x→ −∞

limx→ −∞

(ex +2)=2

Thm. 3.10: Limits at Infinity: If r is a positive rational number and c is any real number, then .

Furthermore, if is defined when x<0, then .

limx→ ∞

cxr =0

xr

limx→ −∞

cxr =0

Ex. 1: Find .limx→ ∞

(3−4x2 +

7x3 )

Ex. 2: Find .limx→ ∞

4x+63x−1

You Try: Find each limit.

a.

b.

c.

limx→ ∞

4x−13x+2

limx→ ∞

4x2 −13x+2

limx→ ∞

4x−13x2 +2

Guidelines for Finding Limits at of Rational Functions: 1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients.3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

±∞

Ex. 3: Find each limit.a.

b.

limx→ ∞

−3x+1

x2 + x

limx→ −∞

−3x+1

x2 + x

You Try: Find each limit.a.

b.

limx→ ∞

x

x2 +1

limx→ −∞

x

x2 +1

Ex. 4: Find .limx→ ∞

x−cosxx

III. infinite limits at infinityDef. of Infinite Limits at Infinity: Let f be a function defined on the interval .1. means that for each positive number M, there is a corresponding number N>0 such that f(x)>M whenever x>N.2. means that for each negative number M, there exists a corresponding number N>0 such that f(x)<M whenever x>N.*This works for , too.

limx→ ∞

f(x) =∞(a,∞)

limx→ ∞

f(x) =−∞

limx→ −∞

Ex. 5: Find .limx→ −∞

(12x−4x2 )

Ex. 6: Find .limx→ ∞

3x2 −2xx+1

You Try: Find .limx→ −∞

−9x3 + 4xx2 + 3