Let’s develop a simple method to find infinite limits and horizontal asymptotes. Here are 3 functions that appear to look almost the same,

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Let’s develop a simple method to find
infinite limits and horizontal asymptotes.
Here are 3 functions that appear to look
almost the same, but there are subtle
differences.
6x  2
y 2
2x  4
6x  2
y 2
2x  4
2
6x  2
y 2
2x  4
3
Let’s explore each as x approaches ∞
lim
x 
6x  2
2
2x  4
Look at the degree of
each polynomial
The degree of the bottom, 2, is greater
than the degree of the top, 1.
As x grows without bound, the bottom
will dominate and the limit will go to 0
lim
x 
6x  2
0
2
2x  4
6x  2
y 2
2x  4
2
Here, the degree of the
top is equal to the degree
of the bottom (both are 2)
The limit will be the ratio of the leading
coefficients (the coefficients of the terms
of highest degree).
6x  2 6


3
2
2x  4 2
2
lim
x 
6x  2
y 2
2x  4
3
Here, the degree of the
top, 3, is greater than the
degree of the bottom, 2.
The numerator will dominate and this limit
will grow without bound to infinity.
6x  2

2
2x  4
3
lim
x 
We can quickly find the horizontal asymptotes:
6x  2
y 2
2x  4
y = 0, same as the limit, this
is the x-axis
6x  2
y 2
2x  4
y = 3, a horizontal line
6x  2
y 2
2x  4
No horizontal asymptote, the
function grows without bound and
does not approach a single value
2
3
Here is a quick quiz for you. Find the
horizontal asymptotes:
5 x 3  7 x  3 The degrees are the
y
3
3x  2 x
same (3) so y = 5/3
4x  6x
y
2
3x  5 x
2
4
The degree of the top is
greater (4 > 2) so there is
no horizontal asymptote

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