Module 11 Lesson 2
Rational Functions Notes Part 2
Although we didn’t technically call them ‘limits’, we partially discussed this topic
earlier in the course. Remember when we talked about these things?


Horizontal Asymptotes
Vertical Asymptotes
Limits as 𝒙 → ±∞
We have talked about this lesson’s topic previously. Think ‘horizontal asymptote’.
Here’s the notation the line 𝑦 = 𝐿 is a horizontal asymptote of a function if either of
the following statements are true:
lim 𝑓(𝑥) = 𝐿
𝑥→∞
𝑜𝑟
lim
𝑥→ −∞
𝑓(𝑥) = 𝐿
This says that f(x) gets arbitrarily
close to some number, L, as ‘x’ gets
infinitely large.
This says that f(x) gets arbitrarily
close to some number, L, as ‘x’ gets
infinitely small.
f(x) gets arbitrarily close to some
number, L, as ‘x’ gets infinitely large.
f(x) gets arbitrarily close to some
number, L, as ‘x’ gets infinitely large.
Infinite Limits as 𝒙 → 𝒂 (where “a” is a real number)
We have talked about this topic as well. Think ‘vertical asymptote’.
Here’s the notation the line 𝑥 = 𝑎 is a vertical asymptote of a function if either of the
following statements are true:
lim 𝑓(𝑥) = ± ∞
𝑥→ 𝑎+
𝑜𝑟
lim 𝑓(𝑥) = ±∞
𝑥→ 𝑎−
This says that as you approach x from
the right-handed side, the y-values of
𝑓(𝑥) gets infinitely large or infinitely
small.
This says that as you approach x from
the left-handed side, the y-values of
𝑓(𝑥) gets infinitely large or infinitely
small.
f(x) gets arbitrarily close to some
number, L, as ‘x’ gets infinitely large.
f(x) gets arbitrarily close to some
number, L, as ‘x’ gets infinitely large.
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Let’s take a look at a rational function with which you are familiar……. 𝑓(𝑥) = 𝑥
Some quick facts about this rational function are:
 The domain is (−∞, 0) & (0, +∞)
 The range is (−∞, 0) & (0, +∞)
 There is a vertical asymptote at 𝑥 = 0
 There is a horizontal asymptote at 𝑦 = 0
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Here is the graph of 𝑓(𝑥) = 𝑥
Example 1:
Now let’s find the limits of the following. Find the limits with a graph and using a
table
a) lim
1
𝑥→∞ 𝑥
=
As your x values
become infinitely
larger, your y values
approach the x-axis
(𝑦 = 0 )
As your x values
become infinitely
larger, your y values
approach the value of 0
(which is the x-axis).
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Therefore lim
𝑥→∞ 𝑥
= 0 . That also means this function has a horizontal
asymptote at 𝑦 = 0
b) lim
1
𝑥→−∞ 𝑥
=
As your x values become
infinitely smaller, your y
values approach the xaxis (𝑦 = 0 )
Therefore lim
1
𝑥→ −∞ 𝑥
As your x values become
infinitely smaller, your y
values approach the
value of 0 (which is the xaxis).
= 0 . That also means this function has a horizontal
asymptote at 𝑦 = 0.
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c) lim+ 𝑥 =
𝑥→0
As you approach the value of x=0 from the right,
the y values get infinitely larger.
Notice that at x=0
there is an
ERROR for the y
value. This
means that the
function is
undefined at x=0.
Remember that 0
is not part of the
domain.
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Therefore lim+ 𝑥 = + ∞ , this function has a vertical asymptote at 𝑥 = 0.
𝑥→ 0
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Notice that at x=0
there is an ERROR for
the y value. This
means that the
function is undefined
at x=0. Remember
that 0 is not part of
the domain.
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d) lim− 𝑥 =
𝑥→ 0
As you approach the
value of x=0 from the
left, the y values get
infinitely smaller.
As you approach the value of x=0
from the left, the y values get
infinitely smaller. (to see the y
values get smaller, change the
TABLSET delta Tbl)
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Therefore lim− 𝑥 = − ∞ , this function has a vertical asymptote at 𝑥 = 0.
𝑥→ 0
Example 2: Find the following limits.
a) lim
1
𝑥→∞ 𝑥−2
+3=
Remember from the rational function
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transformations that if 𝑓(𝑥) = 𝑥 was the
parent function, then this function would
shift right 2 units and up 3 units.
As the x values get infinitely
larger, the y values approach
the value of 3.
Therefore lim
1
𝑥→∞ 𝑥−2
+ 3 = 3 (Verifying that y=3 is a horizontal asymptote.)
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b)
lim
5𝑥 2 +3
𝑥→ −∞ 2𝑥 2 −2
=
As the x values get infinitely smaller the y values approach 2.5. (The
calculator doesn’t have enough room to show all the digits therefore it
rounds the number to 2.5. To prove that these numbers actually do not
equal 2.5 see below. )
Therefore lim
5𝑥 2 +3
𝑥→ −∞ 2𝑥 2 −2
c) lim
5
= 2 (This verifies that y=2.5 is a horizontal asymptote.)
3𝑥 2 −1
𝑥→∞ 𝑥 3 −1
As the x values get infinitely larger the y values approach 0. (The
calculator doesn’t have enough room to show all the digits therefore it
shows the numbers in scientific notation.)
Therefore lim
3𝑥 2 −1
𝑥→∞ 𝑥 3 −1
= 0 (This verifies that 𝑦 = 0 is a horizontal asymptote.)
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d)
lim
3𝑥 2 −4
𝑥→ −∞ 4𝑥 2 +𝑥
As the x values get infinitely smaller the y values approach 0.75. (The
calculator doesn’t have enough room to show all the digits therefore it
rounds the number to 0.75. To prove that these numbers actually do
not equal 0.75 see below.
Therefore lim
3𝑥 2 −4
𝑥→ −∞ 4𝑥 2 +𝑥
3
3
= 4 (This verifies that 𝑦 = 4 is a horizontal asymptote.)
The above examples concentrated on limits as x approached negative or positive
infinity. Now let’s look at some examples where the limit is equal to negative or
positive infinity.
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Example 3: Find the following limits.
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a) lim+ 𝑥−2 + 3 =
𝑥→2
As we approach the value of 2 from the right, the y values become
infinitely larger. In the table notice that the y value when 𝑥 = 2 says
ERROR. This means that 𝑥 = 2 is not in the domain of this function.
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Therefore lim+ 𝑥−2 + 3 = + ∞ (This verifies that 𝑥 = 2 is a vertical asymptote.)
𝑛→2
b) lim−
𝑥→ 3
4𝑥 2 +2𝑥−1
𝑥 2 −9
=
As we approach the value of 3 from the left, the y values become infinitely
smaller. In the table notice that the y value when 𝑥 = 3 says ERROR. This
means that 𝑥 = 3 is not in the domain of this function, this means that
direct substitution could not be used to solve for the limit.
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Therefore lim−
𝑥→ 3
4𝑥 2 +2𝑥−1
𝑥 2 −9
= −∞ (This verifies that 𝑥 = 3 is a vertical asymptote.)
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