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Let's break down the limit step by step:

We are asked to find:

\[
\lim_{{x \to 2^-}} \frac{\sqrt{5x(x - 2)}}{|x - 2|}
\]

### Step 1: Analyze the expression inside the limit.
We notice that the denominator is the absolute value of \((x - 2)\). Since \(x \to 2^-\) means \(x\) is approaching 2 from the left (less than 2), we know that \(x - 2 < 0\). Thus, the absolute value function simplifies to:

\[
|x - 2| = -(x - 2)
\]

So the limit becomes:

\[
\lim_{{x \to 2^-}} \frac{\sqrt{5x(x - 2)}}{-(x - 2)}
\]

### Step 2: Factor out the terms.
Now, notice that we have the same factor \((x - 2)\) in both the numerator (inside the square root) and the denominator. For small values of \(x\) near 2, the expression simplifies as follows:

\[
\frac{\sqrt{5x(x - 2)}}{-(x - 2)} = \frac{\sqrt{(x - 2)} \cdot \sqrt{5x}}{-(x - 2)}
\]

We can cancel out \(\sqrt{(x - 2)}\) with \((x - 2)\), leaving us with:

\[
\lim_{{x \to 2^-}} -\frac{\sqrt{5x}}{\sqrt{x - 2}}
\]

### Step 3: Evaluate the limit.
As \(x \to 2^-\), the term \(\sqrt{5x} \to \sqrt{10}\), and \(\sqrt{x - 2}\) approaches \(0^-\) (because \(x - 2\) is negative for \(x < 2\), but approaching zero). This leads the entire expression to approach negative infinity.

### Step 4: Conclusion.
Since the function blows up negatively as \(x \to 2^-\), the limit does not exist.

Thus, the correct answer is:

**B. Does not exist**

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Would you like further details or explanations on any of these steps?

Here are 5 related questions:
1. How do absolute values affect limit calculations?
2. What is the difference between limits approaching from the left and from the right?
3. Why does the limit blow up to negative infinity in this case?
4. How would the limit change if the square root wasn't involved?
5. How can we graphically interpret limits approaching a point?

**Tip**: When evaluating limits involving absolute values, always consider whether the variable is approaching from the left or right to handle the sign correctly.

Find the limit: lim(x → 2⁻) (√(5x(x - 2)) / |x - 2|).

Learn how to solve the limit lim(x → 2⁻) (√(5x(x - 2)) / |x - 2|). This problem involves key concepts of limits, absolute value, and square roots. We explore the limit’s behavior from the left side and explain step-by-step why the limit does not exist. Suitable for advanced high school or Calculus I students.

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