Problem 64 - Calculus

Question

We know the integral \(\int_{1}^{x} t^{h-1} d t=\left[t^{h} / h\right]_{1}^{x}=\left(x^{h}-1\right) / h\) Its limit as \(h \rightarrow 0\) is__________.

Step-by-Step Solution

Answer

The limit is \(\ln(x)\).

Step 1: Understand the Problem

We are given the integral \(\int_{1}^{x} t^{h-1} dt = \left(t^h / h \right)_{1}^{x} = \frac{x^h - 1}{h}\). We need to find its limit as \(h \rightarrow 0\).

Step 2: Apply the Limit

We have the expression \(\frac{x^h - 1}{h}\) and we need to find \(\lim_{h \to 0} \frac{x^h - 1}{h}\).

Step 3: Recall Logarithmic Limit

We know from calculus, the limit \(\lim_{h \to 0} \frac{x^h - 1}{h} = \ln(x)\). This is a standard result where you recognize the definition of the derivative of \(x^h\) at \(h = 0\), yielding the natural logarithm.

Key Concepts

Limit Evaluation

When dealing with integration and limits, it is vital to evaluate how certain functions behave as variables approach specific values. In this case, we analyze the integral \(\int_{1}^{x} t^{h-1} dt\) as \(h\) approaches 0. Understanding the concept of limit evaluation helps you predict the behavior and outcomes of expressions under this transition.

To evaluate the limit \(\lim_{h \to 0} \frac{x^h - 1}{h}\), we consider how the expression changes as \(h\) gets closer to zero. This involves calculating the value that \(\frac{x^h - 1}{h}\) converges to, simplifying complex expressions, and understanding its defined form. Limit evaluation often requires recognizing patterns and standard results, like a known derivative or logarithmic identity, to successfully predict outcomes.

Calculus

Calculus is a field in mathematics that studies change and motion. It includes the concepts of derivatives, integrals, and limits — all crucial tools for scientists and engineers. In the problem, calculus helps address the challenge of performing integration and evaluating limits to solve complex expressions.

By applying calculus principles, we integrate \(t^{h-1}\) from 1 to \(x\), resulting in the expression \(\left(t^h / h \right)_{1}^{x} = \frac{x^h - 1}{h}\). Calculus not only helps you find the definite integral of a function over an interval but also provides techniques for limit evaluation when variables tend towards certain values. Utilizing calculus techniques like these promotes deeper understanding and easier computation when dealing with diverse mathematical problems.

Logarithmic Limit

The concept of a logarithmic limit is a crucial result in calculus, often arising during derivative calculations. It strengthens your ability to solve certain limits involving exponential functions. Here, we deal with the classic limit:\

\
• \(\lim_{h \to 0} \frac{x^h - 1}{h} = \ln(x)\)
\

This limit represents the derivative of \(x^h\) with respect to a parameter \(h\) evaluated at 0, yielding the natural logarithm \(\ln(x)\).

This result is fundamental for understanding how changes in exponents can relate to logarithms. It highlights the intimate relationship between exponential growth and logarithmic functions in calculus. Grasping this concept allows you to see how logarithms serve as inverses to exponentials and provide crucial insights into the nature of change and growth in mathematical functions.