We are asked to evaluate the following limit:

$\lim_{x \to 3} \left( \frac{|x^2 - 7x|}{x^2 + 1} \right).$

Step 1: Simplify the expression

Let's first simplify each part of the function inside the limit.

• The denominator is $x^2 + 1$, which is a continuous function, and at $x = 3$, this evaluates to: $3^2 + 1 = 9 + 1 = 10.$ Therefore, $x^2 + 1$ is not zero at $x = 3$, and we do not have a problematic denominator.

• The numerator involves the absolute value $|x^2 - 7x|$. Let’s evaluate $x^2 - 7x$ at $x = 3$: $3^2 - 7(3) = 9 - 21 = -12.$ Since the result is negative, the absolute value will negate it: $|x^2 - 7x| = |-12| = 12.$

Step 2: Evaluate the limit

Now we substitute $x = 3$ into the simplified expression:

$\frac{|x^2 - 7x|}{x^2 + 1} = \frac{12}{10} = 1.2.$

Final Answer:

Thus, the value of the limit is:

$\boxed{1.2}.$

Would you like further clarification or more details?

Here are 5 related questions to explore further:

1. How does the absolute value affect the behavior of the function as $x \to 3$?
2. Can you compute the limit if the absolute value were removed from the expression?
3. How does the limit behave if the function were approaching a point where the denominator is 0?
4. Why is it important to check the sign of the expression inside the absolute value before simplifying?
5. Can the limit change if we approach the point from the left-hand side or the right-hand side?

Tip: When dealing with absolute values in limits, always evaluate the expression inside the absolute value first, then decide whether to negate it or not depending on the sign.