For each function, determine the long run behavior:

x^3+1 divided by x^2+1

x^2+1 divided by x^2+2

The long-run behavior of the function as ,   and as ,  .

The long-run behavior of the function as

, .

Step-by-step explanation:

The long-run behavior of rational functions refers to what happens to the graph of the function as the x-values get big or small.

Suppose that is a rational function. Let be the leading term of P(x) and be the leading term of Q(x). Then the long run behavior of f(x) is the same as the long run behavior of the power function

We have the following functions:

(a)

The leading term of the numerator is and the leading term of the denominator is . So, the long run behavior of f(x) is given by

This tells us that when the inputs get extreme (i.e.,) the outputs will be

As , and as ,

(b)

We note that the leading term of the numerator is and the leading term of the denominator is . So, the long run behavior of g(x) is given by

This tells us that when the inputs get extreme (i.e.,) the outputs will be about 1. This means that the graph of g(x) gets closer and closer to the line y = 1 as .

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step-by-step explanation: