Let

f(x)=\begin{cases}\frac{x^{2}-16}{x-4}~~ , if ~~x \neq 4\\kx-24,~~if ~~~ x=4 \end{cases}

## Step By Step Solution:

The given **piecewise****-defined function** is:

f(x)=\begin{cases}\frac{x^{2}-16}{x-4}~~ , if ~~x \neq 4\\kx-24,~~if ~~~ x=4 \end{cases}

Notice that,

f(x) =\frac{x^{2}-16}{x-4}

is not defined at **x=4**.But,

\lim_{x \to 4}f(x)

is defined at **x=4**.Now,

f(x) =kx-24

is defined at **x=4**:

Using the **Continuity of a Function**,

\lim_{x \to a}f(x)=f(a)

\lim_{x \to 4}f(x)=f(4)

Simplify the expression for x=4:

f(x)=kx-24~~, at ~x=4 \\f(4)=4k-24

As we know,

\lim_{x \to 4}f(x)=f(4)

Substitute the

f(x)=\frac{x^{2}-16}{x-4} \\and \\f(4)=4k-24

into the above ,we get:

\lim_{x \to 4}{\left(\frac{x^{2}-16}{x-4}\right)}=4k-24

Evaluate the limit:

\lim_{x \to 4}{\left(\frac{x^{2}-16}{x-4}\right)}=4k-24

\implies\lim_{x \to 4}{\left(\frac{\cancel{(x-4)}(x+4)}{\cancel{x-4}}\right)}=4k-24~~~~~~~~~~~~{\left (a^2-b^2=(a-b)(a+b) \right)}

\implies\lim_{x \to 4}{\left(x+4\right)}=4k-24

Now , putting values as discontinuity is removed:

\implies8=4k-24

\implies4k=32

\therefore k=8