Limits Involving Infinity. Example Problem 1
When doing limits to infinity, the only thing of significance is the degree of the numerator and the degree of the denominator. lim_(x-> infinity) (10x^3 +5x^2 -8)/(3x^6 +9x+2) [Graph] x->infinity y->0 (10x^3)/3x^6 1/infinity^3 -> 0 Horizontal Aymptote of y=0 In this problem, we’re looking at a numerator degree of 3, and the denominator degree is 6. When analyzing this function, we are looking at its behavior as a whole. This function behaves like 10x cubed over 3x to the sixth. Which, when reduced, and the coefficients are effectively negligible, this will behave like 1 over x cubed. When you consider that the graph of 1 over x cubed has a vertical asymptote at 0, it can be seen to be in quadrants III and I. Thus, if we consider the limit of × as × approaches infinity, as x increases without bound, the y-value becomes smaller and smaller. By looking at the equivalent term here, 1 over infinity cubed, you can see that an infinitely large number divided by an infinitely large number results in a very small number. So we’ll say that the limit is 0. We can also say that y = 0 is a horizontal asymptote of the graph
