Box "polynomials, sums and ratios with powers at infinity"

The basic type of expression we will cover here is a linear combination of powers xa with a > 0, exponentials ax with a > 1, general powers xx, and powers of logarithms [ln(x)]a with a > 0. The simplest such expression is a polynomial, a typical example would be

How to find limits at infinity of such expressions:

Step 1. Identify the dominant term of the given expression. First, determine the dominant category (the four kinds listed above) using the scale of powers:

Every type prevails over the types listed later:
(1)   the power xx;
(2)   exponentials ax, a > 1;
(3)   powers xa, a > 0;
(4)   logarithms [ln(n)]a, a > 0.

After determining the dominant category, compare all terms in the given expression that belong to this category (if there are more of them) and choose the dominant among them. Here the rule is simple. Within each category (2), (3), (4), the dominance is given by the constant a, larger constant means dominance.

In this way one determines the dominant term of the whole given expression (there might be more of them, see below).

For practical use people often prefer a more colloquial way of remembering this hierarchy, using phrases like "powers beat logarithms" and "exponentials beat powers" etc. For details, see Intuitive evaluation in Theory - Limits. When determining dominance, we ignore constants by which some powers, exponentials etc. might be multiplied.

If we want to guess the limit of the given expression at infinity and do not care about proper mathematical calculations, we look at the limit of the dominant term, take into account constants by which it might be multiplied in the expression and we have the answer. If we want to confirm this guess by calculations, we go to the next step.

Step 2. If there is a unique dominant term, factor it out and then find the limit of the resulting expression. If there are more dominant terms, they can be put together only if it does not result in disappearance of this dominant term. Then the procedure of factoring out still works.

If there are more dominant terms and putting them together would make the dominant type disappear entirely, then the expression has to be handled differently. Usually one tries to cancel the dominant terms using algebra. The most frequent reason why terms could not be cancelled directly is that some are mixed up in roots. In such cases, try the box "difference of roots".

Sometimes one has to simplify using algebra to identify the types of terms properly.

Example:
Consider the expression above. The last term does not fit the pattern, but we can change it into a proper term:

22x = (22)x = 4x.

Now we see that the expression features powers (x13 and x1/2), exponentials (3x and 4x) and logarithm (note that we ignored multiplicative constants). The dominant category is exponentials. Since we have two of them, we have to decide which one is dominant. The larger base is 4, so the dominant term of the whole expression is 4x. Therefore, for large values of x the given expression behaves like −4x and consequently it has limit negative infinity at infinity.

The fact that the given expression behaves like its dominant term we sometimes write like this:

3x13 − x1/2 + 7⋅3x − [ln(x)]13 − 22x ∼ −4x.

We can say that 4x is the type of the expression as a whole. Mathematically we can confirm this guess by factoring out 4x, see here.

Now we look at more general expressions that can be handled this way.

Roots/powers

Some parts of an expression as above may be under a root or in a power. Example:

Then we follow this procedure:

Step 1. First we determine the type of each individual root/power. For one particular root/power we first determine the dominant type of the expression inside and then apply the root/power to it.

Step 2. We determine the dominant term(s) of the given expression by comparing individual types, but roots/powers are now represented by their types. Then we follow the usual procedure. When doing the proper calculations by factoring out, it is usually easier to first factor out dominant terms from under the roots/powers.

Example: In the expression above we have one root and one composed power. The expression in the root has dominant term x6, after applying the square root we see that the whole root is of the type x3.

The dominant expression inside the composed second power is [ln(x)]2, therefore the power as such is of the type [ln(x)]4.

Now we look at the whole expression, but we imagine that instead of the root there is x3 and instead of the composed power there is [ln(x)]4. There are powers and logarithms in the expression, so the logarithms will get overshadowed by powers and the dominant term will come from among the powers. The larger one is x4, which is the dominant term (and the type) of the whole expression. We can express our reasoning as follows:

We will now show proper calculations, that is, we solve it by factoring out the dominant term:

How did we know that the "infinity over infinity" fractions went to zero? Using the l'Hospital rule. For the second one we can actually use a little trick and the first fraction.

Ratios

Often we encounter fractions where the numerator and denominator are of the above type. Then the procedure is as follows:

Step 1. Find the dominant terms of the numerator and denominator as above.

Step 2. Factor out the dominant terms, then cancel (if possible) and find the limit of their ratio.

Sometimes it is easier to use an alternative, especially if the expressions in the ratio are actually polynomials:

Step 2'. When the dominants of the numerator and denominator are the same, cancel both the numerator and the denominator by the dominant. People actually use cancelling also in other situations, the rule "cancel the smaller dominant" is often quoted, but has a "little" drawback: It often does not work. If you want to find out more about cancelling in ratios, check out this little note.

Since ratios at infinity most often occur with polynomials and related terms, we will try one such example here.

Example: Evaluate (if it exists)

We start with the root in the numerator. The dominant term of the expression inside is x3, so the whole root is of the type x3/2. Since 3/2 < 2, the dominant term in the numerator is x2, which is also the dominant term in the denominator. Since they are the same, instead of factoring out it is easier to cancel them. However, it is probably better to factor the dominant term out of the root first.

In Solved Problems - Limits, these methods are used in this problem, this problem, and this problem. This approach is also used in the second example in the box "difference of roots" and one example in the box "indeterminate ratio".

Next box: polynomials at proper points
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