Thank you for the clarification. Let's rework the problem with this updated information.

Given:

$\lim_{t \to \infty} 1000 \cdot \frac{\frac{1}{999} e^{\frac{\ln \frac{999}{19}}{t} \cdot t}}{1 + \frac{1}{999} e^{\frac{\ln \frac{999}{19}}{t} \cdot t}}$

First, simplify the exponent $\frac{\ln \frac{999}{19}}{t} \cdot t$:

$\frac{\ln \frac{999}{19}}{t} \cdot t = \ln \frac{999}{19}$

So the expression inside the limit simplifies to:

$\lim_{t \to \infty} 1000 \cdot \frac{\frac{1}{999} e^{\ln \frac{999}{19}}}{1 + \frac{1}{999} e^{\ln \frac{999}{19}}}$

Using the property $e^{\ln x} = x$, this becomes:

$\lim_{t \to \infty} 1000 \cdot \frac{\frac{1}{999} \cdot \frac{999}{19}}{1 + \frac{1}{999} \cdot \frac{999}{19}}$

Further simplification gives:

$1000 \cdot \frac{\frac{999}{999 \cdot 19}}{1 + \frac{999}{999 \cdot 19}}$

$1000 \cdot \frac{\frac{1}{19}}{1 + \frac{1}{19}}$

$1000 \cdot \frac{1}{19 + 1}$

$1000 \cdot \frac{1}{20}$

$50$

Therefore, the limit is:

$\lim_{t \to \infty} 1000 \cdot \frac{\frac{1}{999} e^{\frac{\ln \frac{999}{19}}{t} \cdot t}}{1 + \frac{1}{999} e^{\frac{\ln \frac{999}{19}}{t} \cdot t}} = 50$

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. What role does the exponential function play in determining the behavior of the limit as $t \to \infty$?
2. How does changing the ratio inside the logarithm affect the final result of the limit?
3. What other types of functions (e.g., polynomials, trigonometric) can be analyzed using similar limit techniques?
4. How can these types of limits be interpreted in practical applications, such as physics or engineering?
5. What are the common pitfalls to avoid when dealing with limits involving exponential and logarithmic functions?

Tip: When working with limits involving exponential and logarithmic terms, always look for simplifications using properties of logarithms and exponentials, as they can greatly reduce the complexity of the expression.