Question

Suppose f and g are continuous functions such that g(6) = 6 and lim x → 6 [3f(x) + f(x)g(x)] = 45. Find f(6).

Answer 1

Since g(6)=6, and both functions are continuous, we have:


`\lim_(x \to 6) [3f(x)+f(x)g(x)] = 45\\\\\lim_(x \to 6) [3f(x)+6f(x)] = 45\\\\lim_(x \to 6) [9f(x)] = 45\\\\9\cdot lim_(x \to 6) f(x) = 45\\\\lim_(x \to 6) f(x)=5`


if a function is continuous at a point c, then
`lim_(x \to c) f(x)=f(c)`,

that is, in a c ∈ a continuous interval, f(c) and the limit of f as x approaches c are the same.


Thus, since
`lim_(x \to 6) f(x)=5`, f(6) = 5


Answer: 5