AP Calculus AB Exam Question:

Find the limit of the function as x approaches infinity:

lim(x→∞) [(5x^4 + 3x^3 - 2x^2 + x + 1) / (2x^4 + 7x^2 - 9)]

Step-by-Step Solution:

To find the limit as x approaches infinity, we need to examine the behavior of the numerator and denominator as x becomes extremely large.

First, let's divide each term in the numerator and denominator by the highest power of x, which is x^4:

lim(x→∞) [(5x^4/x^4) + (3x^3/x^4) + (-2x^2/x^4) + (x/x^4) + (1/x^4)] / [(2x^4/x^4) + (7x^2/x^4) + (-9/x^4)]

Simplifying this expression, we get:

lim(x→∞) [5 + (3/x) - (2/x^2) + (1/x^3) + (1/x^4)] / [2 + (7/x^2) - (9/x^4)]

As x approaches infinity, any term with a nonzero constant divided by x, x^2, x^3, or x^4 will approach zero. Therefore, we can simplify the expression further:

lim(x→∞) [5 + 0 - 0 + 0 + 0] / [2 + 0 - 0]

lim(x→∞) 5/2

Hence, the limit of the given function as x approaches infinity is 5/2.

Note: The key concept used in this example is that for rational functions, as x approaches infinity, the highest degree term in the numerator and denominator determines the behavior of the function.