Suppose f and g are continuous functions such that $g(6) = 5$ and $\lim_{x\to6} [3f(x) + f(x)g(x)] = 40$. Find $f(6)$.

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How in the world do I solve this? I am completely confused. Can someone please help me out?

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3 Answers

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As f and g are continuous functions,

$$\lim_{x\to 6} f(x)=f(6)\text{ and } \lim_{x\to 6} g(x)=g(6)$$

So, $$\lim_{x\to 6} [3f(x)+f(x)g(x)]=f(6)[3+g(6)] $$

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Replace $g(6)$ by its limit, so work out the solution to $\lim_{x\to 6} \{ 3 f(x)+ f(x) 5 \}=40$.

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The left hand side is $$ [3 f(6)+f(6)\cdot 5] = 8f(6)$$ so $$ f(6) = 40/8 =5$$

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