Math 400 (Section 2.4)
Infinite Limits & Limits at Infinity
1. Graph the function f ( x ) =
indicated limits:
Name ________________________________
1
( x − 2)
2
and find the
lim f ( x ) = ________
x → 2−
lim f ( x ) = ________
x → 2+
lim f ( x ) = ________
x→2
Vertical asymptote: _______________
2. Graph the function g ( x ) =
x
( x + 1)( x − 1)
Horizontal asymptote: __________________
and find
the indicated limits:
lim g ( x ) = ________
x →−1−
lim g ( x ) = ________
x →1−
lim g ( x ) = ________
x →−1+
lim g ( x ) = ________
x →1+
Vertical asymptotes: _____________________
Horizontal asymptote: __________________
lim h ( x ) = _______
2x −1
.
x −1
lim+ h ( x ) = _______
lim h ( x ) = _______
lim h ( x ) = _______
3. Graph the function h ( x ) =
x →∞
x →1−
x →1
x →1
Horizontal asymptote: _____________
Vertical asymptote: ___________
Use limit properties to prove that h(x) approaches the horizontal asymptote as x → ∞ .
4. Graph the function f ( x ) = x3 − 4 x and determine
the indicated limits:
lim f ( x ) = ________ lim f ( x ) = ________
x →∞
x →−∞
Note: these limits are describing the _______
behaviors of the polynomial function f ( x ) .
5. Consider the graph of the function g ( x ) = 2 +
sin ( π2 x )
3
x
below.
This type of graph is often referred to as a dampened sinusoid. What does lim g ( x )
x →∞
appear to be? _______
6. Recall the Squeeze Theorem:
Assume that functions f, g, and h are such that f ( x ) ≤ g ( x ) ≤ h ( x ) for all x close to c, except
possibly at c itself. If lim f ( x ) = lim h ( x ) = L , then lim g ( x ) = L .
x →c
x →c
x →c
We wish to use this theorem the prove that our guess in Problem 5 is correct. To do this, what
must c be equal to? _________

sin ( π x ) 
Now prove that lim  2 + 3 2  = 2.
x →∞
x 
