For the following two (02) items: Consider the function f(x) =...

For the function to be continuous at $x = 0$, the left-hand limit must be equal to the right-hand limit and the value of the function at $x = 0$.

Left-hand limit at $x = 0$ is given by:
$\lim_{x \to 0^{-}} f(x) = \lim_{x \to 0^{-}} 4(5^x) = 4(5^0) = 4$

Right-hand limit at $x = 0$ is given by:
$\lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{+}} (8k + x) = 8k + 0 = 8k$

Equating the left-hand limit and the right-hand limit:
$8k = 4$

$k = \dfrac{4}{8} = 0.5$

Answer: $0.5$