5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 7, 2018 9:52

Limits and Continuity 85

6.1 The Limit of a Function

Main Concepts:Definition and Properties of Limits, Evaluating Limits, One-Sided Limits,

Squeeze Theorem

Definition and Properties of Limits

Definition of Limit

Let f be a function defined on an open interval containinga, except possibly ataitself.

Then limx→af(x)=L(read as the limit of f(x)asxapproachesaisL) if for anyε>0, there

exists aδ>0 such that|f(x)−L|<εwhenever|x−a|<δ.

Properties of Limits

Given limx→a f(x)=Land limx→ag(x)=MandL,M,a,c, andnare real numbers, then:

1. limx→ac=c


2. limx→a[cf(x)]=cxlim→a f(x)=cL


3. limx→a[f(x)±g(x)]=xlim→a f(x)±xlim→ag(x)=L+M


4. limx→a[f(x)·g(x)]=xlim→a f(x)·xlim→ag(x)=L·M


5. limx→a
   f(x)
   g(x)

=

xlim→a f(x)

xlim→ag(x)

=

L

M

,M= 0

6. limx→a[f(x)]n=

(

xlim→a f(x)

)n

=Ln

Evaluating Limits

Iffis a continuous function on an open interval containing the numbera, then limx→af(x)=

f(a).

Common techniques in evaluating limits are:

STRATEGY

1. Substituting directly


2. Factoring and simplifying


3. Multiplying the numerator and denominator of a rational function by the conjugate of
   either the numerator or denominator


4. Using a graph or a table of values of the given function

Example 1

Find the limit: limx→ 5

√

3 x+1.

Substituting directly: limx→ 5

√

3 x+ 1 =

√

3(5)+ 1 =4.