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Let's solve each limit one by one.

### Problem 1:
\[
\lim_{{x \to 0^+}} \sqrt{x} \ln(x)
\]

### Solution:
Rewrite \(\sqrt{x}\) as \(x^{1/2}\), so:
\[
\lim_{{x \to 0^+}} x^{1/2} \ln(x) = \lim_{{x \to 0^+}} \frac{\ln(x)}{x^{-1/2}}
\]
This is an indeterminate form of type \(\frac{-\infty}{\infty}\), so we can apply L'Hôpital's Rule:
\[
= \lim_{{x \to 0^+}} \frac{\frac{d}{dx}(\ln(x))}{\frac{d}{dx}(x^{-1/2})} = \lim_{{x \to 0^+}} \frac{\frac{1}{x}}{-\frac{1}{2}x^{-3/2}}
\]
Simplify:
\[
= \lim_{{x \to 0^+}} \frac{-2}{\sqrt{x}} = -\infty
\]

So,
\[
\lim_{{

Find the limit. Use l'Hôpital's Rule where appropriate. If there is a more elementary method, consider using it.

1. \( \lim_{{x \to 0^+}} \sqrt{x} \ln x \)

2. \( \lim_{{x \to \infty}} x^2 e^x \)

3. \( \lim_{{x \to \infty}} \sqrt{x^2 + x} - x \)

4. \( \lim_{{x \to \pi/2^+}} (\sec x - \cot x) \)

5. \( \lim_{{x \to \infty}} (x^{1/x}) \)

6. \( \lim_{{x \to \infty}} \left( \frac{x}{x + 2} \right)^x \)

Explore solutions to a variety of limit problems, including using L'Hôpital's Rule and other elementary methods. Problems involve limits approaching zero, infinity, and specific angles. Ideal for college calculus students.

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