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The problem requires proving the following limit:

\[
\lim_{x \to 2^+} \ln(x - 2) = -\infty
\]

It also provides the formal definition for limits approaching \(-\infty\):

For any \(M > 0\), there exists a \(\delta > 0\) such that for all \(x\) satisfying \(0 < x - a < \delta\), it holds that \(f(x) < -M\).

### Proof Outline:
We are tasked to prove that for any \( M > 0 \), there exists a \(\delta > 0\) such that for all \( x \) with \(0 < x - 2 < \delta\), we have:

\[
\ln(x - 2) < -M
\]

### Step-by-Step Proof:

1. **Choose \( M > 0 \)**: Let \(M\) be any positive number.
   
2. **Find the corresponding \(\delta > 0\)**: We want to find a \(\delta\) such that whenever \(0 < x - 2 < \delta\), the logarithmic expression \(\ln(x - 2)\) is less than \(-M\).

   - Since \(\ln(x - 2)\) decreases as \(x\) approaches 2 from the right (i.e., as \(x - 2\) gets smaller), we need to find how small \(x - 2\) must be to make \(\ln(x - 2)\) less than \(-M\).
   
   - Recall that for the natural logarithm to satisfy \(\ln(y) < -M\), we must have \(y < e^{-M}\) (i.e., the argument of the logarithm must be smaller than \(e^{-M}\)).

3. **Set \(x - 2 = \delta\)**: From the inequality \(\ln(x - 2) < -M\), we deduce that \(x - 2 < e^{-M}\). Therefore, choose \(\delta = e^{-M}\).

4. **Conclusion**: For this choice of \(\delta = e^{-M}\), whenever \(0 < x - 2 < \delta\), it follows that:

   \[
   \ln(x - 2) < -M
   \]

Thus, we have formally proven that \(\lim_{x \to 2^+} \ln(x - 2) = -\infty\), as required.

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How does the logarithmic function behave as its argument approaches 0 from the positive side?
2. What are the key differences between limits approaching \(+\infty\) and \(-\infty\)?
3. How does the concept of delta-epsilon proofs generalize to other types of limits?
4. What is the significance of choosing \(\delta = e^{-M}\) in this proof?
5. How would the proof change if we were asked to show that the limit approaches \(+\infty\) instead of \(-\infty\)?

**Tip**: When working with logarithmic limits, keep in mind that the argument of the logarithmic function must always be positive!

In Lecture 2, we discussed the example lim ln(x - 2) = -∞. We can now formally prove this equality, and your task is to do so. Note that the equality lim f(x) = -∞ formally means the following: for any M > 0, there exists a δ > 0 such that for all x satisfying 0 < x - a < δ, it holds that f(x) < -M.

This exercise requires proving the limit lim ln(x - 2) = -∞ as x approaches 2 from the right. Learn how to use the epsilon-delta definition of limits to formalize this concept, with detailed step-by-step reasoning.

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