Continuity and Differentiability

Consider the function $f(x) = \frac{\tan(3x) - 3x}{x^3}$ defined for $x \neq 0$. If we define $f(0) = L$ to make $f$ continuous at $x = 0$, find $L$.

Consider the function $f(x) = \frac{x^2 - 9}{x - 3}$ with $f(3) = 6$. Evaluate $\lim_{x \to 3} f(x)$ to determine if $f$ is continuous at $x = 3$.

A biologist models the growth rate of a bacteria colony as $R(t) = \frac{\sqrt[3]{t} - 2}{t-8}$ for $t \neq 8$ hours, and defines $R(8) = \frac{1}{12}$. Determine if $R$ is continuous at $t=8$ by evaluating $\lim_{t \to 8} R(t)$.