Question

if possible, choose k so that the following function is continuous on any interval: $f(x)=\begin{cases}\frac{3x^{5}-12x^{4}}{x - 4}&x
eq4\\k&x = 4end{cases}$ $k=\text{ }$ (if no k will make the function continuous, enter none)

Explanation:

Step1: Simplify the function for $x

eq4$
Factor the numerator of $\frac{3x^{5}-12x^{4}}{x - 4}$. We have $3x^{5}-12x^{4}=3x^{4}(x - 4)$. So, $\frac{3x^{5}-12x^{4}}{x - 4}=\frac{3x^{4}(x - 4)}{x - 4}=3x^{4}$ for $x
eq4$.

Step2: Find the limit as $x$ approaches 4

We find $\lim_{x
ightarrow4}\frac{3x^{5}-12x^{4}}{x - 4}=\lim_{x
ightarrow4}3x^{4}$. Substitute $x = 4$ into $3x^{4}$, we get $3\times4^{4}=3\times256 = 768$.

Step3: Use the continuity condition

For the function $f(x)$ to be continuous at $x = 4$, we need $\lim_{x
ightarrow4}f(x)=f(4)$. Since $\lim_{x
ightarrow4}\frac{3x^{5}-12x^{4}}{x - 4}=768$ and $f(4)=k$, then $k = 768$.

Answer:

$768$