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BCC.01.6 – Limits Involving
Infinity
MCB4U - Santowski
(A) Infinite Limits
• Consider the function f(x) = x-2 and then the lim x0 (x-2)
• If we try a direct substitution, we get (0)-2 which equals 1/02 which is
undefined
• So what is the limit???
• We can try a numeric approach and substitute in numbers close to but
not equal to x = 0+ (like x = 0.001 and 0.000001) and x = 0- (like 0.00001 and -0.000001)
• Then the values of f(x) are 1000, 1000000 for x = 0.001 and 0.000001
and 100000 and 1000000 for x = -0.00001 and -0.000001
• As it turns out, as x 0+ and 0-, the values of f(x) get larger and larger
(f(x)  +∞)
• So we do not reach a limiting number for f(x), meaning that this limit
is undefined
(A) Infinite Limits – Graph of f(x) = x-2
(A) Infinite Limits - Summary
1
lim 2
• Consider x 0 x which we said does not exist because the
values of f(x) do not approach a number; i.e. the function
does not reach a limiting value.
• We are not regarding ∞ as a number, simply as a concept
meaning "increasing without bound" or that the value of f(x)
= x-2 can be made arbitrarily large as we get closer and
closer to x = 0.
1
 
• So we will write this as lim
x 0 x 2
• And what we see on the graph is a vertical asymptote at
x=0
(B) Examples of Infinite Limits
•
2
lim
x 3 x  3
•
lim log( x)
x 0
lim
tan(
x
)
•

x
2
• Work through these
three examples
numerically,
graphically, and
algebraically
(C) Limits at Infinity
• In considering limits at infinity, we are being asked to make
our x values infinitely large and thereby consider the “end
behaviour” of a function 2
• Consider the limit lim x  1 numerically, graphically
2
and algebraically x x  1
• We can generate a table of values and a graph (see next
slide)
• So here the function approaches a limiting value, as we
make our x values sufficiently large  we see that f(x)
approaches a limiting value of 1  in other words, a
horizontal asymptote
(C) Limits at Infinity – Graph & Table
• Table of Values
•
•
•
•
•
•
•
•
x
-10000.0000
-6666.66667
-3333.33333
0.00000
3333.33333
6666.66667
10000.00000
y
1.00000
1.00000
1.00000
-1.00000
1.00000
1.00000
1.00000
(C) Limits at Infinity – Algebra
•
 x2 1

lim 
x    x 2  1 




 lim 
x   


x2
1

x2
x2
x2
1

x2
x2
1 

1


2 
x

 lim 
x   
1 
1 2 
x 

1 0

1 0
1






• Divide through by the
highest power of x
• Simplify
• Substitute x = ∞  1/∞  0
(D) Examples of Limits at Infinity
• Work through the following examples graphically, numerically or
algebraically
• (i)

lim tan
x 
1
x 
 3x 2  x  2 

• (ii) lim 
2
x   5 x  4 x  1


• (iii)
lim
x 
 x  2  x
2
(E) Infinite Limits at Infinity
• Again, recall that in considering limits at infinity, we are
being asked to make our x values infinitely large and
thereby consider the “end behaviour” of a function
• Consider the limit lim x∞ ¼x3 numerically, graphically
and algebraically
• We can generate a table of values and a graph (see next
slide)
• As it turns out, as x +∞ and as x -∞, the values of f(x)
get larger and larger (f(x)  +∞)
• So we do not reach a limiting number for f(x), meaning
that this limit is undefined
(F) Infinite Limits at Infinity –
Graph and Table
• A table of values:
•
x
• -1000
• -600
• -200
• 200
• 600
• 1000
y
-250000000.
-54000000.0
-2000000.00
2000000.000
54000000.00
250000000.0
(G) Examples of Infinite Limits at
Infinity
• Work through the following examples graphically, numerically or
algebraically
•
•

lim x  x
x
2

 x2  x 

lim 
x  3  x


(H) Internet Links
• Limits Involving Infinity from Paul
Dawkins at Lamar University
• Limits Involving Infinity from Visual
Calculus
• Limits at Infinity and Infinite Limits from
Pheng Kim Ving
• Limits and Infinity from SOSMath
(I) Homework
• Handouts from Stewart, 1997, Chap 2.9