Limits at Infinity
Limits at infinity truly are not so difficult once you've become familiarized with
then, but at first, they may seem somewhat obscure. The basic premise of limits
at infinity is that many functions approach a specific y-value as their independent
variable becomes increasingly large or small. We're going to look at a few
different functions as their independent variable approaches infinity, so start a
new worksheet called 04-Limits at Infinity, then recreate the following graph.
In this graph, it is fairly easy to see that as x becomes increasingly large or
increasingly small, the y-value of f(x) becomes very close to zero, though it never
truly does equal zero. When a function's curve suggests an invisible line at a
certain y-value (such as at y=0 in this graph), it is said to have a horizontal
asymptote at that y-value. We can use limits to describe the behavior of the
horizontal asymptote in this graph, as:
and
Try setting xmin as -100 and xmax as 100, and you will see that f(x) becomes
very close to zero indeed when x is very large or very small. Which is what you
should expect, since one divided by a large number will naturally produce a small
result.
The concept of one-sided limits can be applied to the vertical asymptote in this
example, since one can see that as x approaches 3 from the left, the function
approaches negative infinity, and that as x approaches 3 from the right, the
function approaches positive infinity, or:
and
Unfortunately, the behavior of functions as x approaches positive or negative
infinity is not always so easy to describe. If ever you run into a case where you
can't discern a function's behavior at infinity--whether a graph isn't available or
isn't very clear--imagining what sort of values would be produced when tenthousand or one-hundred thousand is substituted for x will normally give you a
good indication of what the function does as x approaches infinity.
Powers and Exceptions
Let's analyze a couple of not-so-straightforward examples concerning limits at
infinity to ensure a full understanding of how they work. Take a look at this
strangely bird-like function:
The graph gives it away; the limit of the function as x approaches either positive
or negative infinity is two. But what if we didn't have the graph? There are actually
a couple of clues as to what value the function approaches at both infinities.
Compare the numerator to the denominator: see how the "highest power" of each
polynomial is 4? The numerator has 2*x4 and the denominator has simply x4.
Perhaps you see where this is going--what do you get when you divide 2*x4 by
x4? Why, 2 of course. Simply put, when the numerator and the denominator of a
function have the same power--when the highest exponents of x between the top
and bottom of the fraction match--the function will have a horizontal asymptote at
y equals (coefficient of numerator's highest power) / (coefficient of denominator's
highest power).
Well...maybe that wasn't so simple. Let me reiterate with a general case. For a
function like this one:
Assuming that n is the highest exponent of both the numerator and denominator
and that a and b are coefficients:
For this trick to work, though, the function must be a rational expression (one
polynomial divided by another) and cannot have trigonometric or other special
functions in the numerator or denominator. For example, the horizontal
asymptote of the following function cannot be found using the above method, as
it refers to an expression where x is an exponent.
Before you graph this function to see if and where it has a horizontal asymptote,
though, let us try a bit of logical analysis. First, which will grow the fastest, e^x or
5^x? Evaluating 'float(e)' in Sage will tell you that Euler's number e equals
approximately 2.7182818284590451, so 5^x will grow more quickly than e^x. But
what about the 2*x in the denominator? Consider the growth, now, of 5^x relative
to 2*x. When x is 5, the former evaluates to 3125 while the latter evaluates to 10.
What you should gather from this is that 5^x trumps all; as x approaches positive
or negative infinity, the function will become the quotient of a big number (e^x)
divided by a really big number that only continues to grow. In other words, one
could write this as:
And tada! There you have it. Comparing relative growths is another way of
figuring out the behavior of a function as its independent variable approaches
infinity. Continue reading to learn about this lesson's final type of function, those
that oscillate.
Oscillating Functions
Like most of the trigonometric functions, as x approaches positive or negative
infinity, the sine function itself continues to jump up and down. An oscillating
function is one that continues to move between two or more values as its
independent variable (x) approaches positive or negative infinity. The limit of an
oscillating function f(x) as x approaches positive or negative infinity is undefined.
In the plot that you just created, try replacing 'sin' with various other trigonometric
functions such as 'cos', 'tan', 'sec', 'csc', 'cot', or any of the arc-functions (like
'arctan') to observe whether they oscillate or whether they have a horizontal
asymptote as x approaches positive or negative infinity.
The next graph behaves differently though, as you may observe:
Even though the function oscillates indefinitely due to the sine function in its
numerator, I can tell you without a doubt that the limit of the function as x
approaches either positive or negative infinity is still zero. But why is this? The
sine of a really big number must still be somewhere in the range of -1 and 1,
while denominator will simply be a really big number. If we create a table of
values, we can watch the function's behavior when x is large. Use the following
code to create such a table. table()
There is another way to prove that the limit of sin(x)/x as x approaches positive or
negative infinity is zero. Whether you have heard of it as the pinching theorem,
the sandwich theorem or the squeeze theorem, as I will refer to it here, the
squeeze theorem says that for three functions g(x), f(x), and h(x),
If
then
and
.
,
To apply this theorem, we have to find a function g(x) that is less than f(x) as well
as a function h(x) that is greater than f(x). Since sin(x) is always somewhere in
the range of -1 and 1, we can set g(x) equal to -1/x and h(x) equal to 1/x. We
know that the limit of both -1/x and 1/x as x approaches either positive or
negative infinity is zero, therefore the limit of sin(x)/x as x approaches either
positive or negative infinity is zero. One could write this out as:
Given that
and
,
since
for all x in
and
,
it follows that
.
Practice Problems
Find the limit of each function as x approaches positive and negative infinity, if
either exists. Try determining each limit by analysis and/or plotting, then check
your results in the Notebook using 'limit(..., x=...)'.
1)
2)
3)