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We explore functions that behave like horizontal lines as the input grows without
Let’s start with an example:
Consider the function: with the table below
What does the table tell us about as grows bigger and bigger?
As grows bigger and bigger, it seems that approaches . It seems that we should
Now consider a different function:
Fill in the table below, rounding to decimal places. What does the table tell us
about as grows bigger and bigger?
As grows bigger and bigger, it seems like
approaches . It seems that we should write
If becomes arbitrarily close to a specific value by making sufficiently large, we
write and we say, the limit at infinity of is .
If becomes arbitrarily close to a specific value by making sufficiently large
and negative, we write and we say, the limit at negative infinity of is
Compute the limit. We will confirm what we guessed earlier by computing the limit
at infinity. We can assume that all the Limit Laws also apply to limits at infinity.
Sometimes one must be careful, consider this example.
In this case we multiply the numerator and denominator by
, which is a positive number as since , is a negative number.
Note, since and we can also apply the Squeeze Theorem when taking limits at
infinity. Here is an example of a limit at infinity that uses the Squeeze Theorem, and
shows that functions can, in fact, cross their horizontal asymptotes.
We can bound our function Now write with me
And we also have
Since we conclude by the Squeeze Theorem, .
If then the line is a horizontal asymptote of .
Give the horizontal asymptotes of
From our previous work, we see that , and upon
further inspection, we see that . Hence the horizontal asymptote of is the line
It is a common misconception that a function cannot cross an asymptote. As the next
example shows, a function can cross a horizontal asymptote, and in the example this
occurs an infinite number of times!
Give a horizontal asymptote of
Again from previous work, we see that . Hence is a
horizontal asymptote of .
We conclude with an infinite limit at infinity.
The function grows very slowly, and seems like it may have a horizontal asymptote,
see the graph above. However, if we consider the definition of the natural log as the
inverse of the exponential function
means that and that is positive.
We see that we may raise to higher and higher values to obtain larger numbers. This
means that is unbounded, and hence .