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

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to 1^+} x^{\frac{1}{1 - x}}\).
2. Recognize that as \(x \to 1^+\), the base \(x\) approaches 1 and the exponent \(\frac{1}{1 - x}\) tends to \(+\infty\), creating an indeterminate form of type \(1^{\infty}\).
3. Rewrite the expression using the exponential and natural logarithm to handle the indeterminate form: \(x^{\frac{1}{1 - x}} = e^{\frac{1}{1 - x} \cdot \ln(x)}\).
4. Focus on finding the limit of the exponent: \(\lim_{x \to 1^+} \frac{\ln(x)}{1 - x}\). This is a \(\frac{0}{0}\) indeterminate form, so apply L'Hôpital's Rule by differentiating numerator and denominator separately.
5. After applying L'Hôpital's Rule, evaluate the resulting limit and then substitute back into the exponential expression to find the original limit.

Key Concepts

Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially when the function approaches a point of discontinuity or an indeterminate form, is essential for analyzing the behavior near that point.

Indeterminate Forms

Indeterminate forms like 0^0, ∞^0, or 1^∞ arise in limit problems where direct substitution does not yield a clear answer. Recognizing these forms helps in applying appropriate techniques such as logarithmic transformation or L'Hôpital's Rule to resolve the limit.

Logarithmic Transformation for Limits

When limits involve expressions raised to variable powers, taking the natural logarithm can simplify the problem by converting the power into a product. This allows the use of limit laws and L'Hôpital's Rule on the exponent, after which exponentiation recovers the original limit.