https://file.aiquickdraw.com/mathbot/file/1726198618566hrrd8r01.jpg

The problem in the image asks to find the limit:

\[
\lim_{\Delta x \to 0^+} \frac{\frac{3}{x + \Delta x} - \frac{3}{x}}{\Delta x}
\]

### Solution:

This is a standard difference quotient used in derivative calculations. We are tasked with simplifying this expression and taking the limit as \(\Delta x \to 0^+\).

1. **Combine the fractions in the numerator:**

\[
\frac{3}{x + \Delta x} - \frac{3}{x} = \frac{3x - 3(x + \Delta x)}{x(x + \Delta x)} = \frac{3x - 3x - 3\Delta x}{x(x + \Delta x)} = \frac{-3\Delta x}{x(x + \Delta x)}
\]

2. **Simplify the entire expression:**

Now substitute this back into the original limit expression:

\[
\lim_{\Delta x \to 0^+} \frac{\frac{-3\Delta x}{x(x + \Delta x)}}{\Delta x}
\]

This simplifies to:

\[
\lim_{\Delta x \to 0^+} \frac{-3\Delta x}{\Delta x \cdot x(x + \Delta x)}
\]

3. **Cancel out \(\Delta x\):**

\[
\lim_{\Delta x \to 0^+} \frac{-3}{x(x + \Delta x)}
\]

4. **Take the limit as \(\Delta x \to 0^+\):**

When \(\Delta x\) approaches 0, \(x + \Delta x \to x\), so the expression becomes:

\[
\frac{-3}{x \cdot x} = \frac{-3}{x^2}
\]

### Final Answer:
\[
\boxed{-\frac{3}{x^2}}
\]

This is the value of the limit.

Do you need any additional clarifications or details? Here are a few related questions:

1. How is this difference quotient related to derivatives?
2. What is the derivative of \(\frac{3}{x}\)?
3. What happens if \(x = 0\) in this limit?
4. Can you explain why we can cancel \(\Delta x\) in the simplification?
5. How would the limit change if the function was \(\frac{5}{x}\) instead of \(\frac{3}{x}\)?

**Tip:** When solving limits with complex fractions, it's often helpful to combine terms and cancel out common factors before applying the limit.

Find the limit (if it exists). If an answer does not exist, enter DNE.

lim (Δx → 0^+) ((3/(x + Δx)) - (3/x)) / Δx

Learn how to find the limit of a difference quotient involving fractions and approach the concept of derivatives in calculus. The solution includes combining fractions, simplifying complex fractions, and applying the limit as Δx approaches 0.

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