Honors Calculus
Limits as x Becomes Infinite
 Evaluate limits as x becomes infinitely large or small.
Limits at infinity truly are not so difficult once you've become familiar with them, 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.
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
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 ten-thousand or onehundred thousand is substituted for x will normally give you a good indication of what the function does as x
approaches infinity.
Powers and Exceptions
Take a look at the function to the right:
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).
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. Let’s consider the following function:
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? Since the number e equals approximately
2.7182818284590451, 5^x will grow more quickly than e^x. But what about the 2x in the denominator?
Consider the growth, now, of 5^x relative to 2x. 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:
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. 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.
Let’s try some problems…
Page 77. 46-50 even, 58, 60, page 85, 33-51 odd