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

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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



  1. 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.
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