To evaluate the limit

$\lim_{x \to 2} \frac{x+5}{|2 - x|} \sin x,$

we'll analyze it carefully.

Step 1: Separate the expressions

We can break the given expression into separate parts:

$f(x) = \frac{x+5}{|2 - x|} \sin x$

where we will handle the behavior of each factor as $x \to 2$.

• The term $x + 5$ is continuous, so as $x \to 2$, we have:

  $\lim_{x \to 2} (x + 5) = 7$

• The term $\sin x$ is also continuous, and as $x \to 2$, we get:

  $\lim_{x \to 2} \sin x = \sin(2) \approx 0.9093$

Step 2: Handle the absolute value

The absolute value function $|2 - x|$ behaves differently depending on whether $x$ approaches 2 from the left ($x \to 2^-$) or from the right ($x \to 2^+$).

• If $x \to 2^+$, then $|2 - x| = x - 2$, which approaches 0 from the positive side.
• If $x \to 2^-$, then $|2 - x| = 2 - x$, which also approaches 0 but from the positive side.

In both cases, $|2 - x| \to 0^+$ as $x \to 2$, and thus the term $\frac{x + 5}{|2 - x|}$ becomes unbounded.

Step 3: Analyze the overall limit

• As $x \to 2$, the term $\frac{x+5}{|2-x|}$ blows up to infinity (approaching $+\infty$).
• Meanwhile, $\sin(2)$ remains finite.

Thus, the entire expression behaves like $\infty \times \text{finite}$, and the limit tends to infinity.

Conclusion:

The limit does not exist in a finite sense; instead, the limit tends to infinity:

$\lim_{x \to 2} \frac{x+5}{|2 - x|} \sin x = +\infty.$

Would you like more details on any step?

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Here are five related questions to extend your understanding:

1. What are the properties of the absolute value function in limits?
2. How does the behavior of $\frac{1}{|2-x|}$ change when $x \to 2$?
3. What happens if we replace $\sin x$ with a different trigonometric function?
4. How do we approach limits involving absolute value functions and discontinuities?
5. Can you evaluate $\lim_{x \to 2} \frac{x+5}{2-x} \sin x$ without the absolute value?

Tip: When evaluating limits involving absolute values, always consider the left- and right-hand behaviors of the function separately!