When considering functions made up of the sums, differences, products or quotients of different sorts of functions (polynomials, exponentials and logarithms), or different powers of the same sort of function we say that one function dominates the other. This means that as x approaches infinity or negative infinity, the graph will eventually look like the dominating function.
- Exponentials dominate polynomials,
- Polynomials dominate logarithms,
- Among exponentials, larger bases dominate smaller,
- Among polynomials, higher powers dominate lower,
For example, consider the function . The exponential function dominates the polynomial. As , the graph looks like an exponential approaching infinity; that is, . As the graph looks like an exponential with very small but still positive values; that is .
Another example, consider a rational function (the quotient of two polynomials). If the numerator is of higher degree than the denominator, as the numerator dominates, and the limit is infinite. If the denominator is of higher degree, the denominator dominates, and the limit is zero. (And if they are of the same degree, then the limit is the ratio of the leading coefficients. Dominance does not apply.)
Dominance works in other ways as well. Consider the graphs of and . In a standard graphing window, the graphs appear to intersect twice. But on the right side the exponential function is lower than the polynomial. Look farther out and farther up, the exponential dominates and will eventually lie above the polynomial (after x = 7.334).
Here’s an example that pretty much has to be done using the dominance approach.
The polynomial function in the denominator, even with the very small exponent, will dominate the logarithm function. The denominator will eventually get larger than the numerator and drive the quotient towards zero. We will return to this function when we know about finding maximums and points of inflection and find where it starts decreasing. For more on this see my post Far Out!