Question

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}c=0\right.$

Short answer

f is not continuous at $c=0$.

Step-by-step solution

Step 1. Given information 

We have been given a function $f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}\right.$.

We have to determine whether this function is continuous at $c=0$.

Step 2. Write the condition for continuity

We say that f is continuous at c if it is defined at c, given that c is in the domain of f so that f(c) is equal to a number.

Also, another condition is when the right and left limits at c of the function f(x) are both equal to f(c).

$\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$

However, in a case of a piecewise-defined function, different functions for different intervals must be taken into consideration.

Step 3. Find f(0).

Since$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}\right.$

we get $f\left(0\right)=-2$.

Step 4. Find left side limit. 

$$\begin{gathered} \underset{x\to 0^{-}}{lim} \left(\frac{x^2-6x}{x^2+6x}\right) \\ =\underset{x\to 0^{-}}{lim} \left(\frac{x(x-6)}{x(x+6)}\right) \\ =\underset{x\to 0^{-}}{lim} \left(\frac{x-6}{x+6}\right) \\ =\frac{\underset{x\to 0^{-}}{lim} x-\underset{x\to 0^{-}}{lim} 6}{\underset{x\to 0^{-}}{lim} x+\underset{x\to 0^{-}}{lim} 6} \\ =\frac{0-6}{0+6} \\ =\frac{-6}{6} \\ =-1 \end{gathered}$$

Step 5. Find right-side limit. 

$$\begin{gathered} \underset{x\to 0^{+}}{lim} \left(\frac{x^2-6x}{x^2+6x}\right) \\ =\underset{x\to 0^{+}}{lim} \left(\frac{x(x-6)}{x(x+6)}\right) \\ =\underset{x\to 0^{+}}{lim} \left(\frac{x-6}{x+6}\right) \\ =\frac{\underset{x\to 0^{+}}{lim} x-\underset{x\to 0^{+}}{lim} 6}{\underset{x\to 0^{+}}{lim} x+\underset{x\to 0^{+}}{lim} 6} \\ =\frac{0-6}{0+6} \\ =\frac{-6}{6} \\ =-1 \end{gathered}$$

$$\begin{gathered} \underset{x\to 0^{-}}{lim} f\left(x\right)=\underset{x\to 0^{+}}{lim} f\left(x\right) \\ \underset{x\to 0}{lim} f\left(x\right)=-1 \end{gathered}$$

Since $\underset{x\to 0}{lim} f\left(x\right)\ne f\left(0\right)$,

The function is not continuous at $c=0$.