Suppose f and g are continuous functions such that g(7) = 6 and lim [3f(x) + f(x)g(x)] = 18.
x → 7
Find f(7).
To find f(7), we need to utilize the given information and apply the concept of limits.
Given that g(7) = 6, it implies that the limit of g(x) as x approaches 7 exists and equals 6.
We are also given that the limit of [3f(x) + f(x)g(x)] as x approaches 7 equals 18.
Using algebraic manipulation, we can rewrite the expression as:
lim (3f(x) + f(x)g(x)) = 18
x → 7
Next, we can factor out f(x) from the expression:
lim f(x) ⋅ (3 + g(x)) = 18
x → 7
Now, we can substitute the given value of g(7) = 6 into the expression:
lim f(x) ⋅ (3 + 6) = 18
x → 7
Simplifying further:
lim f(x) ⋅ 9 = 18
x → 7
Dividing both sides of the equation by 9:
lim f(x) = 2
x → 7
Hence, we have found that the limit of f(x) as x approaches 7 is 2. Therefore, f(7) = 2.