$$ \lim_{x\to2}\frac{x+5}{\left|2-x\right|}\sin x $$

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!

This article explains the step-by-step process to solve the limit lim_{x→2} (x+5)/|2-x| sin(x). The solution includes an analysis of the absolute value, continuity, and behavior of trigonometric functions near discontinuities. Suitable for students in Grades 11-12.

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