The denominator will eventually get larger than the numerator and drive the quotient towards zero. The polynomial function in the denominator, even with the very small exponent, will dominate the logarithm function. Here’s an example that pretty much has to be done using the dominance approach. Look farther out and farther up, the exponential dominates and will eventually lie above the polynomial (after x = 7.334). But on the right side the exponential function is lower than the polynomial. In a standard graphing window, the graphs appear to intersect twice. Dominance does not apply.)ĭominance works in other ways as well. (And if they are of the same degree, then the limit is the ratio of the leading coefficients. If the denominator is of higher degree, the denominator dominates, and the limit is zero. If the numerator is of higher degree than the denominator, as the numerator dominates, and the limit is infinite. Īnother example, consider a rational function (the quotient of two polynomials). As the graph looks like an exponential with very small but still positive values that is. As, the graph looks like an exponential approaching infinity that is. The exponential function dominates the polynomial. Among polynomials, higher powers dominate lower,įor example, consider the function.Among exponentials, larger bases dominate smaller,. This means that as x approaches infinity or negative infinity, the graph will eventually look like the dominating function. 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.
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