Problem 39 - Calculus Single Variable

Question

Use the logarithm to reduce the given limit to one that can be handled with l'Hôpital's Rule. \(\lim _{x \rightarrow+\infty} x^{1 / \sqrt{x}}\)

Step-by-Step Solution

Answer

The limit is 1 as \(x \to +∞\).

Step 1: Recognize the Limit Type

The given limit presents an indeterminate form of the type "+∞ ^ 0" as \(x \to +∞\). To resolve this, we'll first apply the logarithm to the expression \(x^{1/\sqrt{x}}\).

Step 2: Apply the Natural Logarithm

Let's set \(y = x^{1/\sqrt{x}}\). Taking the natural logarithm on both sides gives us \(\ln(y) = \ln(x^{1/\sqrt{x}}) = \frac{1}{\sqrt{x}} \ln(x)\). This step helps to simplify the expression as a product inside the limit.

Step 3: Transform the Limit

We now aim to find \(\lim_{x \to +∞} \ln(y)\), which is the same as \(\lim_{x \to +∞} \frac{\ln(x)}{\sqrt{x}}\). This transformation is useful because it converts the original power-form limit into a rational form amenable to l'Hôpital's Rule for indeterminate expressions, specifically \(\frac{\infty}{\infty}\).

Step 4: Apply l'Hôpital's Rule

Notice that \(\lim_{x \to +∞} \frac{\ln(x)}{\sqrt{x}}\) is an \(\frac{\infty}{\infty}\) form, allowing us to use l'Hôpital's Rule: differentiate the numerator and the denominator with respect to \(x\). We differentiate \(\ln(x)\) to get \(1/x\) and \(\sqrt{x}\) to get \(1/(2\sqrt{x})\).

Step 5: Simplify the Derivatives

We can now simplify: \[ \lim_{x \to +∞} \frac{1/x}{1/(2\sqrt{x})} = \lim_{x \to +∞} \frac{2\sqrt{x}}{x} = \lim_{x \to +∞} \frac{2}{\sqrt{x}} \]. As \(x \to +∞\), \(\sqrt{x} \to +∞\), thus \(\frac{2}{\sqrt{x}} \to 0\).

Step 6: Conclude the Original Limit

Since \(\lim_{x \to +∞} \ln(y) = 0\), we have \(\ln(y) \to 0\), which implies \(y \to e^0 = 1\). Therefore, the original limit \(\lim_{x \to +∞} x^{1/\sqrt{x}} = 1\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a mathematical function that helps simplify complex expressions, especially when faced with exponential or indeterminate forms. By applying the natural logarithm to a power expression, such as \( x^{1/\sqrt{x}} \) in this exercise, we can transform the problem into a more manageable form.

This is achieved by leveraging the property of logarithms that allows the exponent to be brought down as a coefficient: \( \ln(x^a) = a \cdot \ln(x) \). By simplifying the expression like this, complex operations, like dealing with exponents, become regular algebraic manipulations.

• Natural logarithms are base \( e \), where \( e \approx 2.71828 \).
• The natural logarithm is usually more useful than common logarithms in calculus, particularly when dealing with limits.
• This is due to its relationship with derivatives and integrals, which often appear in calculus problems.

Indeterminate Form

In the realm of calculus, an indeterminate form can be thought of as a mathematical expression where substitution of limits does not directly result in a clear outcome. Common types include expressions like \( \frac{0}{0} \) or \( \infty - \infty \).

In this specific exercise, we deal with the indeterminate form \( +\infty^0 \). This arises when attempting to find the limit \( \lim_{x \to +\infty} x^{1/\sqrt{x}} \). A direct substitution leads to ambiguity because both the base \( x \to +\infty \) and the exponent \( 1/\sqrt{x} \to 0 \) independently behave in extreme opposite manners.

• To resolve this, we use the natural logarithm to convert the exponential expression into a form more amenable to other calculus techniques like l'Hôpital's Rule.
• Once the form is transformed into \( \frac{\infty}{\infty} \), l'Hôpital's Rule can be applied to further simplify the expression.
• This method works because l'Hôpital's Rule specifically addresses these forms by employing derivatives.

Limit Transformation

Limit transformation refers to the process of converting a complex limit into a simpler or different form that is easier to evaluate. This is crucial when dealing with exponential functions or expressions leading to indeterminate forms.

In our exercise, we initially transform the limit \( \lim_{x \to +\infty} x^{1/\sqrt{x}} \) using a natural logarithm. By doing so, the limit becomes \( \lim_{x \to +\infty} \frac{\ln(x)}{\sqrt{x}} \), which is a \( \frac{\infty}{\infty} \) form. This specific transformation allows us to use l'Hôpital's Rule, making it essential in resolving the limit.

• Limit transformation typically involves applying logarithms, trigonometric identities, or algebraic manipulation.
• This helps convert expressions into rational forms when faced with powers, products, or quotient expressions.
• Once in a simpler form, calculus techniques like l'Hôpital's Rule or substitution can be applied efficiently.

The key takeaway here is recognizing when and how to transform limits to make them manageable and solvable using elementary calculus principles.