Problem 56
Question
Find the limits $$\lim _{x \rightarrow \infty} x^{1 / \ln x}$$
Step-by-Step Solution
Verified Answer
The limit is \(e\).
1Step 1: Understanding the Limit Expression
We need to find the limit of the expression \(x^{1 / \ln x}\) as \(x\) approaches infinity. This is an indeterminate form \(\infty^0\), which suggests we need to use logarithmic transformation to evaluate it.
2Step 2: Take the Natural Logarithm
Take the natural logarithm of the expression to simplify it: \[y = x^{1 / \ln x} \quad \Rightarrow \quad \ln y = \frac{1}{\ln x} \cdot \ln x = 1\]This simplifies to \(\ln y = 1\).
3Step 3: Exponential of Both Sides
Since \(\ln y = 1\), take the exponential of both sides to solve for \(y\): \[ y = e^1 = e\] Thus, the original expression \(x^{1 / \ln x}\) converges to \(e\) as \(x\) approaches infinity.
Key Concepts
Indeterminate FormsLogarithmic TransformationExponential Functions
Indeterminate Forms
In calculus, indeterminate forms often arise when trying to evaluate limits. These forms, such as \(0/0\), \(\infty/\infty\), or \(\infty^0\), signify an expression where direct substitution doesn't easily yield a determinable limit.
Recognizing these forms is crucial as they require specific techniques, like L'Hôpital's Rule or algebraic manipulation, to resolve. In our exercise, the expression \(x^{1/\ln x}\) results in the indeterminate form \(\infty^0\) as \(x\) approaches infinity.
The base \(x\) heads towards infinity and the exponent \(\frac{1}{\ln x}\) heads towards zero, leading to this indeterminacy. Such forms hint that a transformation or rethinking is necessary to find the limit.
Recognizing these forms is crucial as they require specific techniques, like L'Hôpital's Rule or algebraic manipulation, to resolve. In our exercise, the expression \(x^{1/\ln x}\) results in the indeterminate form \(\infty^0\) as \(x\) approaches infinity.
The base \(x\) heads towards infinity and the exponent \(\frac{1}{\ln x}\) heads towards zero, leading to this indeterminacy. Such forms hint that a transformation or rethinking is necessary to find the limit.
Logarithmic Transformation
Logarithmic transformation can simplify complex limit expressions, especially those involving powers. It employs the properties of logarithms to conditionally restructure and later solve the expression.
For the expression \(x^{1/\ln x}\), taking the natural logarithm helps by converting the problem of exponential growth into one of linear growth. The transformation is as follows:
Therefore, logarithmic transformation resolves the indeterminate form and props us up to solve the limit.
For the expression \(x^{1/\ln x}\), taking the natural logarithm helps by converting the problem of exponential growth into one of linear growth. The transformation is as follows:
- Set the expression equal to a variable, say \(y = x^{1/\ln x}\).
- Take the natural log of both sides: \(\ln y = \ln(x^{1/\ln x})\).
- Using properties of logarithms, simplify to \(\ln y = \frac{1}{\ln x} \cdot \ln x = 1\).
Therefore, logarithmic transformation resolves the indeterminate form and props us up to solve the limit.
Exponential Functions
Exponential functions, characterized by their rapid growth or decay, often emerge in calculus while resolving limits, derivatives, and integrals.
In our problem, after transforming the expression using logs, we reached \(\ln y = 1\). To return to the function of \(y\), we exponentiate both sides of the equation. This helps to solve for the original expression:
Utilizing exponential functions makes solving limits straightforward once the expression has been simplified by other means, like logarithmic transformation.
In our problem, after transforming the expression using logs, we reached \(\ln y = 1\). To return to the function of \(y\), we exponentiate both sides of the equation. This helps to solve for the original expression:
- From \(\ln y = 1\), you take the exponential of both sides: \(y = e^1\).
- This simplifies to \(y = e\), a constant value, as \(x\) approaches infinity.
Utilizing exponential functions makes solving limits straightforward once the expression has been simplified by other means, like logarithmic transformation.
Other exercises in this chapter
Problem 55
Evaluate the integrals. $$\int \frac{3 d r}{\sqrt{1-4(r-1)^{2}}}$$
View solution Problem 55
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{x(x+1)}$$
View solution Problem 56
Evaluate the integrals. $$\int_{0}^{\pi / 2} 2 \sinh (\sin \theta) \cos \theta d \theta$$
View solution Problem 56
Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{-x}$$
View solution