SOLUTION: Let f be the function defined by f(x)=(cx-(5x^2))/((2x^2)+ax+b), where a, b, and c are constants. The graph of f has a vertical asymptotes at x=1, and f has a removable discontinui

SOLUTION: Let f be the function defined by f(x)=(cx-(5x^2))/((2x^2)+ax+b), where a, b, and c are constants. The graph of f has a vertical asymptotes at x=1, and f has a removable discontinui

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Question 1164742: Let f be the function defined by f(x)=(cx-(5x^2))/((2x^2)+ax+b), where a, b, and c are constants. The graph of f has a vertical asymptotes at x=1, and f has a removable discontinuity at x=-2.
a) show that a=2 and b=-4
b) find the value of c, justify your answer.
d) Write an equation for the horizontal asymptote to the graph of f. Please show work.
c) In order to make f continuous at x=-2, f(-2) should be defined as what? Justify the answer

Thank you!!!!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

If f has a vertical asymptote at

then

must be a factor of

and if f has a removable discontinuity at

then

must be a factor of both the numerator and the denominator functions.

So we need another factor of 2 for the denominator so the denominator function is

Demonstrating that

and

Since

is also a factor of

, if you factor

from the numerator function you get

, so

means that

The lead coefficient of the numerator is -5 and the lead coefficient of the denominator is 2, therefore the equation of the horizontal asymptote is

For the last part, remove the

factor from both the numerator and denominator and then evaluate at -2

John

My calculator said it, I believe it, that settles it

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