ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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Activity A4 – Finding Limits at Infinity Numerically
So far we have focused on the behavior of functions near specific values of x. Now we’ll
explore what happens to the function’s values when x approaches positive and negative
infinity. That is, lim f ( x) and lim f ( x) .
x 
x 
To find lim( x3  2 x  1) we need to look at the behavior of the function’s values when
x 
the x values become large. To do this, we’ll use the limit applet(RS1) to graph the
function and then use the table at the bottom of the limit applet to help us determine the
limit of the function.
To graph f ( x)  x3  2 x  1 on (, ) , input the values shown below into the limit
applet and then click on Update Graph.
By looking at the graph of f(x), we see the function’s values approach negative infinity as
the x values become large. We can confirm this observation by using the table at the
bottom of the limit applet. To do this, input the function and the x values shown below.
Now, click on Fill Table to see the function’s values.
Thus, lim( x3  2 x  1)   .
x 
1. Use the limit applet to find lim ( x3  2 x  1)
x 
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
Page 2 of 2
2x2  x 1
, use the limit applet to find the following.
3x 2
a) lim f ( x)
2. Given f ( x) 
x 
b) lim f ( x)
x 
x 1
, use the limit applet to find the following.
x2  4
a) lim f ( x)
3. Given f ( x) 
x 
b) lim f ( x)
x 
x3  2 x 2  5
, use the limit applet to find the following.
x2  9
a) lim f ( x)
4. Given f ( x) 
x 
b) lim f ( x)
x 
In mathematics, a horizontal asymptote describes the behavior of a function at its extreme
ends. That is, horizontal asymptotes describe the behavior of a function as x → ∞ and as
x → –∞. Thus, when we find lim f ( x) and lim f ( x) , we are finding the horizontal
x 
x 
asymptotes of a function. A summary(RS2) of the relationship between limits at infinity
and horizontal asymptotes is given at right.
Use the table at the bottom of the limit applet(RS1) to find the horizontal asymptotes of
the following functions.
5. f ( x)   x 2  2 x  1
6. f ( x) 
3x 2  x  1
4x2  1
7. f ( x) 
x 1
x x2
8. f ( x) 
x 2  3x  2
x 1
3