Indeterminate Powers and Products
Find the limits in Exercises 53–68.
55. lim (x → ∞) (ln x)^(1/x)

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Identify the limit expression: \(\lim_{x \to \infty} (\ln x)^{\frac{1}{x}}\).

Recognize that the expression is of the form \(f(x)^{g(x)}\) where both the base and the exponent depend on \(x\). To handle limits of this form, it is often helpful to rewrite the expression using the exponential and logarithm functions.

Rewrite the expression as \(e^{\ln\left((\ln x)^{\frac{1}{x}}\right)} = e^{\frac{1}{x} \ln(\ln x)}\).

Focus on finding the limit of the exponent: \(\lim_{x \to \infty} \frac{\ln(\ln x)}{x}\). Analyze the behavior of the numerator and denominator as \(x\) approaches infinity.

Since \(\ln(\ln x)\) grows very slowly compared to \(x\), the fraction \(\frac{\ln(\ln x)}{x}\) approaches zero. Therefore, the original limit becomes \(e^0\), which you can interpret to find the final limit.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how functions like logarithms and powers behave as x approaches infinity is crucial for evaluating such limits.

Indeterminate Forms

Indeterminate forms occur when a limit expression does not directly yield a clear value, such as 1^∞, 0^0, or ∞^0. Recognizing these forms helps in applying appropriate techniques like logarithms or L'Hôpital's Rule to find the limit.

Logarithmic Transformation for Limits

Taking the natural logarithm of a function can simplify limits involving powers by converting exponentiation into multiplication. This transformation often makes it easier to apply limit laws and L'Hôpital's Rule to evaluate complex expressions.

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.55. lim (x → ∞) (ln x)^(1/x)