Answer the following questions about the function , given that a, b, c, d, e, and f are all unique, positive real numbers:

     (a) At what values of x = h, if any, does ?

     (b) Evaluate .

     (c) Does exist? If so, what is it? If not, why doesn't it?

Difficulty:

Solution

(a) This asks "When will g increase or decrease without bound," and the answer is: Whenever it has a vertical asymptote. You can tell that g has a vertical asymptote whenever a value plugged in for x results in a 0 in the denominator but not the numerator. Thus, the answer is whenever x is equal to f or b. If x = c, you get 0 in both parts of the fraction, so that's no good (it usually means removable discontinuity).

(b) Evaluate this limit using the factoring method. Start by canceling the common factor of in the numerator and denominator and then plug in c for all the x's:

(c) The limit at infinity of a rational function only exists if the degree of the denominator is greater than or equal to the degree in the numerator. If the denominator has a greater degree, the limit will be 0. However, if the degrees are equal (as they are in this case, since the degrees of both the numerator and denominator are 3), the limit will be equal to the leading coefficient of the numerator divided by the leading coefficient of the denominator. In case you don't know, the leading coefficient is the number in front of the variable containing the highest exponent. You don't have to, but if you multiply out the top and bottom, it's really easy to find the leading coefficients, although the multiplying itself is no fun at all:

Clearly, the quotient of the leading coefficients is , and that's the answer. Hopefully, you could tell that without all that multiplying. Obviously, the first term of the numerator and denominator, if multiplied out, will be (none of the x's have any other coefficients to make it more complicated than that). You'd then multiply that by a in the top and e in the bottom.