AP Calculus AB
2.2 Limits Involving Infinity
1.
2.
Name__________________________________
What does  represent?
1

Find lim  2   .
x 
x

Find the limit both based on transformations of its function family and using limit properties.
Based on the work for problem (2), do our limit properties hold for limits involving infinity?
Definition: The line y  b is a horizontal asymptote of the graph of a function y  f  x  if either
lim f  x   b or
x 
3.
Graphically determine lim
x 
x
x2  1
(sketch the graph below). Verify your answer using a table.
The Sandwich Theorem also holds for limits as x   .
4.
lim f  x   b
x
Use the Sandwich Theorem to find lim
x 
sin x
x
5.
5 x  sin x
x 
x
Find lim
Exploration
Even though our properties of limits still hold when x   , we must be careful when applying the properties.
Explore below:
a.
Let f  x   5x  sin x and g  x   x . Find lim f  x  and lim g  x 
b.
Find lim
c.
Given your answer to problem (5), explain the meaning of (a) and (b)
d.
Let h  x   sin 2 x and k  x   cos2 x . Find lim f  x  , lim g  x  and lim h  x   k  x 
e.
Find lim  sin 2 x  cos 2 x  . Does this answer make sense given your answer to part (c)?
x 
x 
f  x
g  x
x 
x 
x 
x 
x 
Infinite Limits as x  a
6.
Determine lim
x0
1
x
7.
Determine lim
x0
1
x
Definition: The line x  a is a vertical asymptote of the graph of a function y  f  x  if either
lim f  x    or lim f  x   
x a
x a
Note: The graph of a quotient does NOT always have a vertical asymptote when the denominator equals 0.
End Behavior Models
The graphs of f1  x   2 x4  3x3  10 x2  5x  4 and f 2  x   2 x4 are shown on two different windows.
What is the graph on the right telling us about our two functions?
Definition: End Behavior Model
8.
3x 6  6 x3  2 x  1
.
4 x3  7 x 2  8 x  1
Use the end behavior model to find lim f  x 
Find an end behavior model for f  x  
x 
9.
3x3  2 x  1
4 x3  7 x 2  8 x  1
Use the end behavior model to find lim g  x 
Find an end behavior model for g  x  
x 
Notes: Limits as x   for rational functions; r  x  
p  x
, with p  x  , q  x  both polynomials
q  x
10.
Let f  x   x  e x . Show that g  x   x is a right end behavior model for f while h  x   e x is a left
end behavior model for f.
AP Exam Practice
2x2  1
lim
is
x   2  x  2  x 
2 x
lim x is
x  2
(A) 4
(A) 1
(B) 2
(B) 1
(C) 1
(C) 0
(D) 2
(D) 
(E) nonexistent
(E) none of these