2. INFINITE LIMITS; VERTICAL AND
HORIZONTAL ASYMPTOTES;
SQUEEZE THEOREM
OBJECTIVES:
•define infinite limits;
•illustrate the infinite limits ; and
•use the theorems to evaluate the limits of
functions.
•determine vertical and horizontal asymptotes
•define squeeze theorem
3. DEFINITION: INFINITE LIMITS
Sometimes one-sided or two-sided limits fail to exist
because the value of the function increase or
decrease without bound.
For example, consider the behavior of for
values of x near 0. It is evident from the table and
graph in Fig 1.1.15 that as x values are taken closer
and closer to 0 from the right, the values of
are positive and increase without bound; and as
x-values are taken closer and closer to 0 from the
left, the values of are negative and
decrease without bound.
x
1
)x(f =
x
1
)x(f =
x
1
)x(f =
4. In symbols, we write
−∞=+∞=
−→→ +
x
1
limand
x
1
lim
0x0x
Note:
The symbols here are not real
numbers; they simply describe particular ways in
which the limits fail to exist. Thus it is incorrect to
write .
∞−∞+ and
( ) ( ) 0=∞+−∞+
20. Determine the horizontal and vertical asymptote of
the function and sketch the graph.( )
3
2
f x
x
=
−
a. Vertical Asymptote:
Equate the denominator
to zero to solve for the
vertical asymptote.
2x02x =⇒=−
Evaluate the limit as x
approaches 2
2
3 3 3
lim
2 2 2 0x x→
= = = ∞
− −
b. Horizontal Asymptote:
Divide both the numerator
and the denominator by the
highest power of x to solve for
the horizontal asymptote.
21. 3 3
0
lim 0
2 2 1 01
x
x
x
x x
→+∞
+∞= = =
−− −
+∞
3 3
0
lim 0
2 2 1 01
x
x
x
x x
→−∞
−∞= = =
−− −
−∞
∴
( )
erceptintxnoistheretherefore
30;
2x
3
0,0)xf(If
2
3
20
3
xf,0xIf
:Intercepts
−
≠
−
==
−=
−
==
.asymptotehorizontalais0,Thus
26. LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE
The Squeeze Principle is used on limit problems where the
usual algebraic methods (factoring, conjugation, algebraic
manipulation, etc.) are not effective. However, it requires
that you be able to ``squeeze'' your problem in between
two other ``simpler'' functions whose limits are easily
computable and equal. The use of the Squeeze Principle
requires accurate analysis, algebra skills, and careful use
of inequalities. The method of squeezing is used to prove
that f(x)→L as x→c by “trapping or squeezing” f between
two functions, g and h, whose limits as x→c are known
with certainty to be L.
