A function g(x) has the following characteristics:

1) It is rational with a quadratic polynomial in both the numerator and the denominator.

2) It has a removable discontinuity at x=1 and the limg(x) as x approaches 1 is -1/9

3) The graph of g(x) has a vertical asymptote at x=5

4) The graph of g(x) has a single x-intercept at x=2

Use this information to find:

1. g(x)
2. g(0)
3. Limg(x) as x approaches infinity

To find the function g(x) with the given characteristics, we can start by analyzing each piece of information and then combining them.

1) The given function is rational with quadratic polynomials in both the numerator and the denominator. We can represent this as:

g(x) = (ax^2 + bx + c) / (dx^2 + ex + f)

2) There is a removable discontinuity at x = 1. This means that the function g(x) has a factor of (x - 1) in both its numerator and denominator, which we can cancel out:

g(x) = ((x - 1)(ax + b)) / ((x - 1)(dx + e))

Canceling out (x - 1):

g(x) = (ax + b) / (dx + e)

3) The limit of g(x) as x approaches 1 is -1/9:

lim(x->1) g(x) = -1/9

Substituting x = 1 into g(x):

g(1) = (-1/9) * (d + e)

4) The graph of g(x) has a vertical asymptote at x = 5. This means that x - 5 is a factor in the denominator:

g(x) = (ax + b) / ((x - 5)(dx + e))

5) The graph of g(x) has a single x-intercept at x = 2. This means that (x - 2) is a factor in the numerator:

g(x) = ((x - 2)(ax + b)) / ((x - 5)(dx + e))

Now, let's solve the remaining unknowns:

2. To find g(0), substitute x = 0 into the function:

g(0) = ((0 - 2)(a(0) + b)) / ((0 - 5)(d(0) + e))
g(0) = (-2b) / (-5e)
g(0) = 2b / 5e

3. To find the limit of g(x) as x approaches infinity:

lim(x->∞) g(x) = lim(x->∞) ((x - 2)(ax + b)) / ((x - 5)(dx + e))

To determine this limit, we need to analyze the highest degree terms. The highest degree terms in both the numerator and the denominator are x^2. So, divide every term by x^2:

lim(x->∞) ((1 - (2/x))(a + bx/x^2)) / ((1 - (5/x))(d + ex/x^2))

As x approaches infinity, the terms (2/x) and (5/x) approach zero:

lim(x->∞) (a + bx/x^2) / (d + ex/x^2)

Since the limits of (bx/x^2) and (ex/x^2) both approach zero, the limit becomes:

lim(x->∞) (a / d)

Therefore, the limit of g(x) as x approaches infinity is a/d.

To summarize:

1. g(x) = ((x - 2)(ax + b)) / ((x - 5)(dx + e))
2. g(0) = 2b / 5e
3. lim(x->∞) g(x) = a/d

Now, with specific values of a, b, d, and e, you can evaluate g(x), g(0), and lim(x->∞) g(x).