Calculus/Infinite Limits

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Infinite Limits

Informal Infinite LimitsEdit

Another kind of limit involves looking at what happens to   as   gets very big. For example, consider the function   . As   gets very big,   gets very small. In fact,   gets closer and closer to 0 the bigger   gets. Without limits it is very difficult to talk about this fact, because   can keep getting bigger and bigger and   never actually gets to 0; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does   have to be to be sure that   is really going towards 0?

In this case, we want to say that, however close we want   to get to 0, for   big enough   is guaranteed to get that close. So we have yet another definition.

Definition: (Definition of a limit at infinity)

We call   the limit of   as   approaches infinity if   becomes arbitrarily close to   whenever   is sufficiently large.

When this holds we write

 

or

 

Similarly, we call   the limit of   as   approaches negative infinity if   becomes arbitrarily close to   whenever   is sufficiently negative.

When this holds we write

 

or

 

So, in this case, we write:

 

and say "The limit, as   approaches infinity, equals   ," or "as   approaches infinity, the function approaches 0.

We can also write:

 

because making   very negative also forces   to be close to   .

Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.

Limits at Infinity of Rational FunctionsEdit

One special case that comes up frequently is when we want to find the limit at   (or  ) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example,   is a rational function. Also, any polynomial is a rational function, since   is just a (very simple) polynomial, so we can write the function   as   , the quotient of two polynomials.

Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.

There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.

  • If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both   and  ) is the ratio of the coefficients of the highest terms.
  • If the numerator has the highest term, then the fraction is called "top-heavy". If, when you divide the numerator by the denominator the resulting exponent on the variable is even, then the limit (at both   and  ) is   . If it is odd, then the limit at   is   , and the limit at   is   .
  • If the denominator has the highest term, then the fraction is called "bottom-heavy" and the limit at both   is 0.

Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as   . Also, a straight power of   , like   , has coefficient 1, since it is the same as   .

ExamplesEdit

Example 1

Find   .

The function   is the quotient of two polynomials,   and   . By our rule we look for the term with highest exponent in the numerator; it's   . The term with highest exponent in the denominator is also   . So, the limit is the ratio of their coefficients. Since  , both coefficients are 1,   .

Example 2

Find  .

Using L'Hôpital's rule  

and

 

We look at the terms with the highest exponents; for the numerator, it is   , while for the denominator it is   . Since the exponent on the numerator is higher, we know the limit at   will be   . So,

  .
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Infinite Limits