which of the following functions has a horizontal asymptote at y=1/2 Solution If

Question

which of the following functions has a horizontal asymptote at y=1/2

Explanation / Answer

If the graph of an equation y = f(x) has a horizontal asymptote y = k where k is some number, that implies that as x increases without bound, f(x) = k, or as x decreases without bound, f(x) = k. If you look at your function, you have f(x) = 3 - (x + b)/(x - c). As x increases without bound, this function approaches g(x) = 3 - x/x = 3 - 1 = 2 (The additive constants become irrelevant for very large x, so the denominator gets very close to just being x). Similarly, as x decreases without bound, f(x) approaches the same constant (replace each instance of x with -x and analyze as if x is increasing without bound). If you are unsatisfied with the hand-waving argument that "as x gets very large, the constant becomes irrelevant", you can prove the following statements about rational functions rigorously. Each one leads to the other: 1. As x increases without bound, the function f(x) = 1/x^k approaches 0. You can prove this any way you like, but to do it with full rigor, we mean that for any difference d between f(x) and 0, we can find a number N such that for all x > N, the difference between f(x) and 0 is strictly less than d. Draw a picture to see what this definition is all about.) 2. If P(x) is a polynomial of degree m and Q(x) is a polynomial of degree n, then as x increases without bound, the rational function f(x) = P(x)/Q(x) behaves in one of 3 ways: a) When m n, f(x) increases or decreases without bound. Once you have done the hard work of proving statement (1), (2) can be done by algebra using (1). Working through these will give you an easy way to decide the asymptotic behavior of any rational function from a glance.

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