ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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Activity B4 – Finding Limits at Infinity Algebraically
In activity A4 we used the table on the limit applet to find lim f ( x) and lim f ( x) . Now
x 
x 
we will discuss how to find lim f ( x) and lim f ( x) algebraically.
x 
x 
1
1
and lim p , where p is a real number
p
x

x
x
strictly greater than zero. Once we know the value of these two limits we can find the
limits of many other functions.
Finding limits at infinity relies on finding lim
x 
1. Use the table at the bottom of the limit applet(RS2) to find the following.
1
a) lim
x  x
1
b) lim
x  x
1
c) lim 5
x  x
1
d) lim 5
x  x
The results from exercise 1 lead to the theorem(RS3) shown at right. We can use the
results of this theorem to find the limits of other functions. Three examples (example
1(RS4), example 2(RS5), example 3(RS6)) are shown at right.
Notice the beginning process for each example is the same. When finding infinite limits
of a rational function:
1) Always divide every term in the function by the highest power of x.
2) Simplify the expression
3) Use limit laws to find the limit of the simplified expression.
Find the limits of the following functions.
5x4  7 x 1
lim
2.
x  7 x 6  x 4  x 3
5 x 2  x  1
x 
2 x2  3
3. lim
4x4  x2  x  1
x 
x3
4. lim
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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 x3  x  5
x  3 x 3  x 2  x
5. lim
When finding limits at infinity of rational functions (i.e. horizontal asymptotes of rational
functions), the limit can be determined by analyzing the degree of the polynomial in the
numerator of the rational function and the degree of the polynomial in the denominator of
the rational function. The results of the previous examples(RS 7) lead to the following
conclusions(RS8). These conclusions can be used as short cuts to finding limits at
infinity of rational functions.
6 x3  4 x  1
.
4 x 2  3x3
a) lim f ( x)  2 because the degree of the polynomial in the numerator is the same as the
6. Determine which of the following is true regarding f ( x) 
x 
degree of the polynomial in the denominator. Therefore the limit as x → ∞ is equal to the
ratio of the two polynomials’ leading coefficients.
b) lim f ( x)  0 because the degree of the polynomial in the numerator is less than the
x 
degree of the polynomial in the denominator.
c) lim f ( x)   because the degree of the polynomial in the numerator is greater than the
x 
degree of the polynomial in the denominator.
d) lim f ( x)   because the degree of the polynomial in the numerator is greater than
x 
the degree of the polynomial in the denominator.
e) lim f ( x)  3 / 2 because the degree of the polynomial in the numerator is the same as
x 
the degree of the polynomial in the denominator. Therefore the limit as x → ∞ is equal to
the ratio of the two polynomials’ leading coefficients.
7 x5  4 x 2
.
5 x8  3 x 4  3
a) lim f ( x)  7 / 5 because the degree of the polynomial in the numerator is the same as
7. Determine which of the following is true regarding f ( x) 
x 
the degree of the polynomial in the denominator. Therefore the limit as x → ∞ is equal to
the ratio of the two polynomials’ leading coefficients.
b) lim f ( x)  0 because the degree of the polynomial in the numerator is less than the
x 
degree of the polynomial in the denominator.
c) lim f ( x)   because the degree of the polynomial in the numerator is greater than the
x 
degree of the polynomial in the denominator.
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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d) lim f ( x)   because the degree of the polynomial in the numerator is greater than
x 
the degree of the polynomial in the denominator.
e) lim f ( x)  4 / 3 because the degree of the polynomial in the numerator is the same as
x 
the degree of the polynomial in the denominator. Therefore the limit as x → ∞ is equal to
the ratio of the two polynomials’ leading coefficients.
x2  2 x7  5
.
x4
a) lim f ( x)  1 because the degree of the polynomial in the numerator is the same as the
8. Determine which of the following is true regarding f ( x) 
x 
degree of the polynomial in the denominator. Therefore the limit as x → ∞ is equal to the
ratio of the two polynomials’ leading coefficients.
b) lim f ( x)  0 because the degree of the polynomial in the numerator is less than the
x 
degree of the polynomial in the denominator.
c) lim f ( x)   because the degree of the polynomial in the numerator is greater than the
x 
degree of the polynomial in the denominator.
d) lim f ( x)   because the degree of the polynomial in the numerator is greater than
x 
the degree of the polynomial in the denominator.
e) lim f ( x)  DNE because the degree of the polynomial in the numerator is not written
x 
in descending order.