(4) Redefine the function $$ f(x)=\frac{x^{2}-4 x-12}{x-6} $$ to be continuous at $$ x=6 $$

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To redefine the function $$ f(x) = \frac{x^{2} - 4x - 12}{x - 6} $$ to be continuous at $$ x = 6 $$, we need to first determine the limit of $$ f(x) $$ as $$ x $$ approaches 6. If this limit exists, we can redefine $$ f(6) $$ to be equal to this limit.

### Step 1: Factor the numerator
The first step is to factor the numerator $$ x^{2} - 4x - 12 $$.

To factor $$ x^{2} - 4x - 12 $$, we look for two numbers that multiply to $$-12$$ and add to $$-4$$. These numbers are $$-6$$ and $$2$$. Thus, we can factor the expression as follows:

$$
x^{2} - 4x - 12 = (x - 6)(x + 2)
$$

### Step 2: Simplify the function
Now we can rewrite the function $$ f(x) $$:

$$
f(x) = \frac{(x - 6)(x + 2)}{x - 6}
$$

For $$ x \neq 6 $$, we can simplify this to:

$$
f(x) = x + 2
$$

### Step 3: Find the limit as $$ x $$ approaches 6
Next, we find the limit of $$ f(x) $$ as $$ x $$ approaches 6:

$$
\lim_{x \to 6} f(x) = \lim_{x \to 6} (x + 2) = 6 + 2 = 8
$$

### Step 4: Redefine $$ f(6) $$
To make the function continuous at $$ x = 6 $$, we redefine $$ f(6) $$ to be equal to the limit we just calculated:

$$
f(6) = 8
$$

### Conclusion
The redefined function is:

$$
f(x) = 
\begin{cases} 
\frac{x^{2} - 4x - 12}{x - 6} & \text{if } x \neq 6 \
8 & \text{if } x = 6 
\end{cases}
$$

This ensures that $$ f(x) $$ is continuous at $$ x = 6 $$.
To make the function $$ f(x) = \frac{x^{2} - 4x - 12}{x - 6} $$ continuous at $$ x = 6 $$, redefine $$ f(6) $$ as 8.

(4) Redefine the function \( f(x)=\frac{x^{2}-4 x-12}{x-6} \) to be continuous at \( x=6 \)

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