https://file.aiquickdraw.com/mathbot/file/1726750095887amhvoru8.jpg

We are tasked with evaluating the limit:

\[
\lim_{{x \to 1}} \ln\left( \frac{x}{e^{8x}} \right)
\]

### Step 1: Simplify the argument of the logarithm

First, rewrite the expression inside the logarithm:

\[
\ln\left( \frac{x}{e^{8x}} \right) = \ln(x) - \ln(e^{8x})
\]

Since \(\ln(e^{8x}) = 8x\) (by the logarithmic identity \(\ln(e^a) = a\)):

\[
\ln(x) - 8x
\]

Thus, the limit becomes:

\[
\lim_{{x \to 1}} \left( \ln(x) - 8x \right)
\]

### Step 2: Evaluate the limit

Now, evaluate each term at \(x = 1\):

- \(\lim_{{x \to 1}} \ln(x) = \ln(1) = 0\)
- \(\lim_{{x \to 1}} 8x = 8(1) = 8\)

Therefore:

\[
\lim_{{x \to 1}} \left( \ln(x) - 8x \right) = 0 - 8 = -8
\]

### Conclusion:

The limit is:

\[
\boxed{-8}
\]

Would you like further details on this solution or have any questions?

### Follow-up questions:
1. How does the natural logarithmic function behave as \(x \to 0\) or \(x \to \infty\)?
2. Can you explain the properties of logarithms that were used in the simplification process?
3. What is the significance of exponential growth in the expression \(e^{8x}\)?
4. How would this problem change if the exponent of \(e\) was different?
5. How do logarithms and exponentials interact in calculus and limits?

### Tip:
When simplifying logarithmic expressions, always remember to apply logarithmic identities like \(\ln(a^b) = b \ln(a)\) and \(\ln(ab) = \ln(a) + \ln(b)\).

Discover how to evaluate the limit of ln(x/e^{8x}) as x approaches 1. This solution covers logarithmic and exponential properties, limit evaluation techniques, and essential calculus concepts. Suitable for Grade 12 or undergraduate calculus students.

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