1978 AP calculus AB Free Response..Please Help?

Question

66. Given the function f defined by f(x) =(2x-2)/(x^2+x-2)

a. For what values of x is f(x) discontinuous?

b. At each point of discontinuity found in part a, determine whether f(x) has a limit and, if so, give the value of the limit.

c. Write an equation for each vertical and horizontal asymptote to the graph of f. Justify your answer.

d. A rational function g(x) =a/(b + x)

is such that g(x) = f(x) wherever f is defined. Find the value of a and b.

Answer 1

For these problems, the first step is to factor the rational function.

f(x) = 2(x-1) / ( (x+2)(x-1) )

a) f(x) is discontinuous whenever the denominator is equal to zero. x=-2, 1.

From here on out, we want to deal with a simplified form.  That simplified form is precisely the function g in part (d), so I'll do that first.

g(x) = 2 / (x+2)

Note that this was found from f(x) by canceling the common factor (x-1).

b) g(x) [and therefore f(x)] has a limit at x=1.  The limit is g(1) = 2/3.

The function doesn't have a limit at x=-2.

c) We can find the vertical asymptote/ horizontal asymptote from g(x) also.  The vertical asymptote is at the discontinuity of g(x): x=-2.  The horizontal asymptote is [easily, through whatever method you know] seen to be y=0.

Answer 2

a) f(x) =(2x-2)/(x^2+x-2)

=(2x-2)/[(x-1)(x+2)]

=2(x-1)/[(x-1)(x+2)]

=2/(x+2)

f(x) is undefined at x= -2 ,1

with an asymptote at x= -2

and a removable discontinuity at x= 1

b)lim 2/(x+2)  = - / (- * -) = -infinity

x-->-2 from the left

lim 2/(x+2)  = - / (- * +) = +infinity

x-->-2 from the right

lim 2/(x+2)  = dne

x-->-2

lim 2/(x+2)  = 2/3

x-->1

c)x=-2 , y=0 b/c

lim 2/(x+2) = 0

x-->infinity

d)2/(x+2) = a/(b + x)

a=2;b=2