Finding the Value of a Continuous Function Given a Limit

We are given that f and g are continuous functions, with f(9) = 6. We also know that the limit of (2f(x) - g(x)) as x approaches 9 is 9. Our goal is to find g(9).

Here's how we can solve this using the properties of limits and continuity:

1. Use the limit definition: Since f and g are continuous, we can use the property that the limit of a sum (or difference) is the sum (or difference) of the limits. This lets us rewrite the given limit expression:

   limx→9 (2f(x) - g(x)) = limx→9 2f(x) - limx→9 g(x)

2. Apply the limit and known values: We know the overall limit is 9 and f(9) = 6. Substituting these values gives:

   9 = 2(6) - limx→9 g(x)

3. Solve for the unknown limit: Simplifying the equation allows us to isolate the limit we need:

   9 = 12 - limx→9 g(x) limx→9 g(x) = 12 - 9 limx→9 g(x) = 3

4. Use continuity to find g(9): Because g(x) is continuous, the limit as x approaches 9 is equal to the function's value at 9:

   g(9) = limx→9 g(x) = 3

Therefore, g(9) is equal to 3.