91Ó°ÊÓ

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits of indeterminate forms. When faced with an expression like \(\frac{0}{0}\) or \(\frac{\text{infinity}}{\text{infinity}}\), we can differentiate the numerator and the denominator separately, then take the limit of the resulting fraction. This allows us to simplify complex limits.
In the given exercise, we encounter the indeterminate form \(\frac{\text{-infinity}}{\text{infinity}}\) when evaluating \(h(x) = \frac{2 \text{ln}(x)}{\frac{1}{x}}\). By applying L'Hôpital’s Rule, we differentiate the numerator, \(2 \text{ln}(x)\), and the denominator, \(\frac{1}{x}\), to transform it into a simpler expression that we can evaluate as \(x\) approaches 0.

Logarithmic differentiation is a method used to take derivatives of functions of the form \(g(x) = x^{f(x)}\) by employing logarithms. This technique simplifies the differentiation of more complicated expressions by transforming them into a more manageable form.
In the problem, we start by expressing \(g(x) = x^{2x}\) in natural logarithmic terms. By defining \(h(x) = \text{ln}(g(x))\), we apply the property \(\text{ln}(a^b) = b \text{ln}(a)\) to rewrite \(g(x)\) as \(\text{ln}(x^{2x}) = 2x \text{ln}(x)\). This simplification is crucial for applying L'Hôpital's Rule later.

The exponential function, denoted as \(e^x\), is a mathematical function that grows rapidly and appears frequently in calculus and complex numbers. It is the inverse function of the natural logarithm and has unique properties that make it useful in various fields.
In the context of the exercise, after computing \(\text{lim}_{x \rightarrow 0^{+}} h(x)\), we need to switch back to the original function \(g(x)\). Since \(h(x) = \text{ln}(g(x))\), we use the exponential function to solve for \(g(x)\). The limit, \(\text{lim}_{x \rightarrow 0^{+}} g(x) = e^{h(x)}\), evaluates to \(e^0 = 1\).

Indeterminate forms are expressions in calculus where it is not immediately clear what the limit is. Typical examples include \(\frac{0}{0}\), \(0^0\), \(\frac{\text{infinity}}{\text{infinity}}\), and others.
In the exercise, \(g(x) = x^{2x}\) creates an indeterminate form of \(0^0\) as \(x\) approaches 0 from the positive direction. By converting this into a limit problem with natural logarithms, we can use L'Hôpital’s Rule to resolve the indeterminate form and find the limit.

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two main branches, differential calculus, concerning slopes of curves (derivatives), and integral calculus, concerning the area under curves (integrals).
This exercise showcases some core concepts of calculus: using logarithmic differentiation to simplify complex functions and applying L'Hôpital's Rule to find limits of indeterminate forms. Mastering these techniques is essential for advancing in more sophisticated areas of mathematics and its applications in science and engineering.