Math Problem Statement
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. \( \lim_{{x \to \infty}} \left( \frac{x}{x + 2} \right)^x \)
Solution
Let's solve each limit one by one.
Problem 1:
Solution:
Rewrite as , so: This is an indeterminate form of type , so we can apply L'Hôpital's Rule: Simplify:
So, [ \lim_{{
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Math Problem Analysis
Mathematical Concepts
Limits
Indeterminate Forms
L'Hôpital's Rule
Formulas
\( \lim_{{x \to a}} \frac{f(x)}{g(x)} \) using L'Hôpital's Rule if \( \frac{0}{0} \) or \( \frac{\pm \infty}{\pm \infty} \) form
Exponential limits \( \lim_{{x \to \infty}} \left( 1 + \frac{1}{x} \right)^x = e \)
Theorems
L'Hôpital's Rule
Limit laws
Indeterminate forms
Suitable Grade Level
College Calculus