https://file.aiquickdraw.com/mathbot/file/172665695201962l8a0kc.jpg

The mathematical expression shown in the image is:

\[
\lim_{{x \to +\infty}} \frac{{(1 + x)^x}}{{2^{x^x}}}
\]

We need to find the limit as \( x \to \infty \).

### Step-by-step Solution:

1. **Analyze the numerator:**
   The numerator is \( (1 + x)^x \). As \( x \) approaches infinity, \( (1 + x)^x \) grows very rapidly. Using logarithmic properties, we know that:

   \[
   (1 + x)^x = e^{x \ln(1 + x)}
   \]

   When \( x \to \infty \), \( \ln(1 + x) \sim \ln(x) \), so the growth of \( (1 + x)^x \) is approximately exponential.

2. **Analyze the denominator:**
   The denominator is \( 2^{x^x} \). As \( x \) approaches infinity, this term grows much faster than \( (1 + x)^x \). This is because \( x^x \) increases significantly faster than \( x \), and raising 2 to the power of \( x^x \) leads to an even more extreme growth.

3. **Compare growth rates:**
   The growth rate of \( (1 + x)^x \) is dwarfed by the growth rate of \( 2^{x^x} \). Thus, as \( x \to \infty \), the fraction \( \frac{{(1 + x)^x}}{{2^{x^x}}} \) tends to 0.

### Conclusion:

\[
\lim_{{x \to +\infty}} \frac{{(1 + x)^x}}{{2^{x^x}}} = 0
\]

Let me know if you would like further details or have any questions.

Here are 5 related questions you might find interesting:
1. What is the growth rate of \( x^x \) compared to \( x! \) as \( x \to \infty \)?
2. How does logarithmic growth compare to polynomial growth at infinity?
3. What is the limit of \( \frac{{(1 + x)^x}}{{e^x}} \) as \( x \to \infty \)?
4. Can L'Hôpital's rule be applied to this problem?
5. How would this limit change if the base of the exponent in the denominator were different?

**Tip:** When comparing limits at infinity, identifying which part of the expression grows fastest is key to determining the overall limit.

Find the limit of (1 + x)^x / 2^(x^x) as x approaches infinity.

This problem explores the limit of (1 + x)^x divided by 2^(x^x) as x tends to infinity. Learn how exponential growth of the numerator and denominator leads to a final limit of zero, using logarithmic properties and growth rate comparison.

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