EVALUATING LIMITS ANALYTICALLY (1.3)EVALUATING LIMITS ANALYTICALLY (1.3)

September 20th, 2012September 20th, 2012

I. Properties of limitsThm. 1.1: Some Basic LimitsLet b and c be real numbers and n be a positive integer.1) lim

x→ cb=b

2) limx→ c

x=c

3) limx→ c

xn =cn

Ex. 1: Evaluate each limit.a. lim

x→ 3x

b. limx→ −54

c. limx→ 6

x2

Thm. 1.2: Properties of Limits:Let b and c be real numbers, n be a positive integer, and f and g be functions with the following limits.

limx→ c

f(x) =L limx→ c

g(x) =K

1) Scalar multiple: 2) Sum or difference: 3) Product:

limx→ c

[b⋅f(x)] =bL

limx→ c

[ f (x)±g(x)] =L±K

limx→ c

[ f(x)⋅g(x)] =LK

4) Quotient: 5) Power:

limx→ c

f(x)g(x)

=LK

,K ≠0

limx→ c

[ f(x)]n =Ln

Ex. 2: Evaluate limx→ 32x3 −4x+9

Thm. 1.3: Limits of Polynomial & Rational Functions:If p is a polynomial function and c is a real number, then .If r is a rational function given byand c is a real number such that , then .

limx→ c

p(x) =p(c)

limx→ c

r(x) =p(x)q(x)

q(c)≠0

limx→ c

r(x) =r(c) =p(c)q(c)

Ex. 3: Find the limit:limx→ −1

3x2 −2x+5x+2

Thm. 1.4: The Limit of a Function Involving a Radical:Let n be a positive integer.

limx→ c

xn = cn

This limit is valid for all values of c if n is odd, but is only valid for c>0 if n is even.

Thm. 1.5: The Limit of a Composite Function:If f and g are functions such that and , then

limx→ c

g(x) =L

limx→ L

f(x) = f(L)

limx→ c

f(g(x)) = f(limx→ c

g(x)) = f(L)

Ex. 4: Find each limit.a.

b.

limx→1

x2 +64

limx→ −2

x3 + x3

Thm. 1.6: Limits of Trigonometric Functions:Let c be a real number in the domain of the given trigonometric function.1) 2)3) 4)5) 6)

limx→ c

sinx=sinc limx→ c

cosx=cosc

limx→ c

tanx=tanc limx→ c

cotx=cotc

limx→ c

cscx=cscc limx→ c

secx=secc

Ex. 5: Find each limit.a.b.c.

limx→ π

tanx

limx→ 2π

x⋅cosx

limx→ 0

cos2 x

Summary: Direct substitution can be used to evaluate limits of all polynomial functions, rational functions, radical functions in which the limits are valid, trigonometric functions, and compositions of each of the aforementioned functions.

II. A STrategy for finding limits

Thm. 1.7: Functions that Agree on All but One Point (Functions with a Hole):Let c be a real number and f(x)=g(x) for all in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and .

x ≠c

limx→ c

f(x) =limx→ c

g(x)

A. dividing out technique

Ex. 6: Find the limit: limx→1

x2 + 4x−5x−1

B. Rationalizing technique

Ex. 7: Find the limit: limx→ 0

x+2 − 2x

A strategy for finding limits

1) Recognize which limits can be evaluated by direct substitution.2) If the limit as x approaches c cannot be evaluated by direct substitution, try the dividing out or rationalizing technique to apply theorem 1.7.3) Use a graph or table to reinforce your conclusion.

III. The squeeze theorem

Thm. 1.8: The Squeeze Theorem:If for all x in an open interval containing c, except possibly at c itself, and if ,then exists and is equal to L.

h(x)≤ f (x) ≤g(x)

limx→ c

h(x) =L =limx→ c

g(x)

limx→ c

f(x)

Thm. 1.9: Two Special Trigonometric Limits:1. 2.lim

x→ 0

sinxx

=1 limx→ 0

1−cosxx

=0

Ex. 8: Use the Squeeze Theorem to find .

limx→ 0

f(x)

9−x2 ≤ f(x) ≤9 + x2

Ex. 9: Find each limit.a.

b.

limx→ 0

1−cos3xx

limx→ 0

1xcotx