Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}c=0\right.$

We have been given a function $f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}\right.$.

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

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.

Since$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-6x}{x^2+6x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\ne 0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x=0\end{array}\right.$

we get $f\left(0\right)=-1$.

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

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

localid="1647085301117" $$\begin{gathered} \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right) \\ \underset{x\to 0}{\lim }f\left(x\right)=-1 \end{gathered}$$

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

The function is continuous at $c=0$.