50 STEP 4. Review the Knowledge You Need to Score High
5.1 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→a f(x)=L(read as the limit off(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:
- xlim→ac=c
- xlim→a[cf(x)]=climx→a f(x)=cL
- xlim→a[f(x)±g(x)]=limx→a f(x)±xlim→ag(x)=L+M
- xlim→a[f(x)·g(x)]=limx→a f(x)·limx→ag(x)=L·M
- xlim→a
f(x)
g(x)
=
limx→a f(x)
limx→ag(x)
=
L
M
,M/= 0
- xlim→a[f(x)]n=
(
limx→a f(x)
)n
=Ln
Evaluating Limits
Iffis a continuous function on an open interval containing the numbera, then limx→a f(x)=
f(a).
Common techniques in evaluating limits are:
STRATEGY
- Substituting directly
- Factoring and simplifying
- Multiplying the numerator and denominator of a rational function by the conjugate of
either the numerator or denominator - 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.