The image is in Estonian and translates to:

"It is known that $\lim_{x \to \infty} f(x) = \infty$. Find $\lim_{x \to \infty} (f(x))^{-f(x)}$."

Let's analyze and solve the limit.

Given: $\lim_{x \to \infty} f(x) = \infty$

We need to find: $\lim_{x \to \infty} (f(x))^{-f(x)}$

Rewrite $(f(x))^{-f(x)}$ as: $(f(x))^{-f(x)} = e^{-f(x) \ln(f(x))}$

So, our problem becomes finding: $\lim_{x \to \infty} e^{-f(x) \ln(f(x))}$

Since $\lim_{x \to \infty} f(x) = \infty$, it implies that $f(x) \ln(f(x)) \to \infty$ as $x \to \infty$. Therefore, the expression $-f(x) \ln(f(x))$ approaches $-\infty$.

Thus: $\lim_{x \to \infty} e^{-f(x) \ln(f(x))} = e^{-\infty} = 0$

Conclusion:

$\lim_{x \to \infty} (f(x))^{-f(x)} = 0$

Would you like more details or have any questions?

Here are 5 questions to explore this further:

1. How do we handle limits when functions approach infinity at different rates?
2. What other techniques can be used to simplify expressions involving $f(x)^{-f(x)}$?
3. What if $f(x)$ approaches infinity at a slower rate? How would that affect the limit?
4. Can this method be applied to limits involving other forms like $f(x)^{g(x)}$?
5. Why does exponential decay (e.g., $e^{-\infty}$) lead to zero?

Tip: Converting expressions to exponential form can often simplify limits involving complex powers.