Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 9

Suppose $f$ and $g$ are continuous functions with $f(3) = 5 $ and $\lim \limits_{x \to 3} [2 f(x) - g(x)] = 4$ , find $g(3)$



Based from the theorem, if $f$ is continuous at number a

$\lim \limits_{x \to a} f(x) = f(a)$




$
\begin{equation}
\begin{aligned}
& \text{Therefore}\\
& \phantom{x} & & \lim \limits_{x \to 3} f(x) = f(3) = 5\\
& \phantom{x} & & \lim \limits_{x \to 3} [2f(x) - g(x)] = 4\\
& \phantom{x} & & 2 \lim \limits_{x \to 3} f(x) - \lim \limits_{x \to 3} g(x) = 4\\
& \phantom{x} & & \lim \limits_{x \to 3} g(x) = 2(5) - 4 = 10 - 4 = 6\\
& \text{Again, from the definition}\\
& \phantom{x} & & \lim \limits_{x \to 3} g(x) = g(3) = 6\\
& \text{Hence, }\\
& \phantom{x}& & g(3) = 6
\end{aligned}
\end{equation}
$