Suppose f and g are continuous functions such that g(4) = 4 and lim x → 4 [3f(x) + f(x)g(x)] = 35. Find f(4).

To find f(4), we need to use the given information and the properties of limits.

We know that the limit of a sum is the sum of the limits, and the limit of a product is the product of the limits, provided that the limits exist.

So, we can write the limit as follows:

lim x → 4 [3f(x) + f(x)g(x)] = lim x → 4 3f(x) + lim x → 4 f(x)g(x)

We know that this limit equals 35, so we have:

35 = lim x → 4 3f(x) + lim x → 4 f(x)g(x)

We can factor out the f(x) from the second limit:

35 = 3 * lim x → 4 f(x) + lim x → 4 [f(x) * g(x)]

We know that g(4) = 4, and the limit of a constant times a function is the constant times the limit of the function, so we can simplify further:

35 = 3 * lim x → 4 f(x) + 4 * lim x → 4 f(x)

Combine like terms:

35 = 7 * lim x → 4 f(x)

Finally, solve for the limit, which is the value of f(4):

f(4) = 35 / 7 = 5

So, f(4) = 5.