limit as x approaches 3 from the positive side of (ln(x^2 -9))

To evaluate the limit

\[
\lim_{{x \to 3^+}} \ln(x^2 - 9),
\]

let's examine the behavior of the function \( f(x) = \ln(x^2 - 9) \) as \( x \to 3^+ \).

### Step 1: Substitute into \( x^2 - 9 \)
As \( x \to 3^+ \), \( x^2 - 9 \) approaches \( 3^2 - 9 = 0 \) from the positive side (since \( x^2 - 9 = (x + 3)(x - 3) \), and as \( x \to 3^+ \), \( x - 3 \to 0^+ \)).

### Step 2: Evaluate \( \ln(x^2 - 9) \) as \( x \to 3^+ \)
Since \( x^2 - 9 \to 0^+ \), we are actually considering \( \ln(y) \) as \( y \to 0^+ \). The natural logarithm function \( \ln(y) \) tends to \( -\infty \) as \( y \to 0^+ \).

### Conclusion
Thus,

\[
\lim_{{x \to 3^+}} \ln(x^2 - 9) = -\infty.
\]

The limit diverges to \( -\infty \) as \( x \) approaches 3 from the right.

Would you like a detailed breakdown of the logarithmic behavior, or do you have other questions?

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1. What are common limits involving logarithmic functions?
2. How does approaching from the positive side affect limits?
3. Why does the limit go to \( -\infty \) instead of \( +\infty \)?
4. How do we evaluate limits approaching zero in logarithmic expressions?
5. What other functions have similar behavior near critical points?

**Tip:** Remember, the logarithmic function \( \ln(x) \) approaches \( -\infty \) as \( x \to 0^+ \).

Explore how to evaluate the limit of ln(x^2 - 9) as x approaches 3 from the positive side. Understand the behavior of logarithmic functions near zero and why this limit diverges to negative infinity. Suitable for advanced high school students studying calculus.

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