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Chapter 8: Problem 10

Evaluate \(L^{\prime}(1)\) by using the sequence \((1+1 / n)\) and the fact that \(e=\lim \left((1+1 / n)^{n}\right)\).

Short Answer

Expert verified
The derivative \(L^{\prime}(1)\) evaluated at \(x = 1\) with using the given sequence and the mathematical constant \(e\) is \(e\).

Step by step solution

01

Define the Function

Define the function \(L(x) = ln(x)\). Notice that \(L^{\prime}(x) = 1/x\).
02

Use the Limit Definition of Derivative

To evaluate \(L^{\prime}(1)\), use the limit definition of the derivative:\[L^{\prime}(1) = \lim_{h \to 0} \frac{L(1 + h) - L(1)}{h}\]
03

Substitute the function into the limit

Let's substitute \(L(x) = ln(x)\) into the limit:\[L^{\prime}(1) = \lim_{h \to 0} \frac{\ln(1 + h) - \ln(1)}{h}\]
04

Simplify the expression in the limit

The natural logarithm of 1 is zero, so simplify the expression to:\[L^{\prime}(1) = \lim_{h \to 0} \frac{\ln(1 + h)}{h}\]
05

Use the fact \[e=\lim \left((1+1 / n)^{n}\right)\]

Notice the limit in question has a similar resemblance to the given expression for \(e\). To make it identical, rewrite \(h\) as \(1/n\), therefore as \(n \to \infty\), \(1/n\) tends to zero which is the same as \(h \to 0\). So we can write the limit as:\[L^{\prime}(1) = \lim_{n \to \infty} \frac{\ln(1 + 1/n)}{1/n}\] and this has the same form of the limit that defines \(e\), except for the natural logarithm function in the numerator.
06

Apply the fact \[e=\lim \left((1+1 / n)^{n}\right)\]

Recall the exact definition of the number \(e\) as the limit of the sequence \((1+1 / n)^{n}\). Rewrite the limit \((1+1 / n)^{n}\) as \(\exp(n \ln (1 + 1/n))\). By the continuity of the exponential, we can exchange limit and function: \(\exp(\lim_{n \to \infty} \{ n \ln (1 + 1/n) \})\). Comparing with the limit in Step 5, we can then conclude that the limit is \(\exp(1) = e\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics focused on change. It primarily deals with two concepts: differentiation and integration. Differentiation is about finding the rate of change of a quantity, while integration focuses on the accumulation of quantities. In our problem, we use differentiation to evaluate the derivative of the function \( L(x) = \ln(x) \) at a specific point.
  • Differentiation: In simple terms, differentiation helps us find how a function changes at any point. For the function \( L(x) = \ln(x) \), its derivative is \( L'(x) = \frac{1}{x} \).
  • Limit Definition of a Derivative: This concept helps us find the derivative at a particular point. It is given by the expression: \[ L^{\prime}(a) = \lim_{h \to 0} \frac{L(a + h) - L(a)}{h} \]
We used this formula to find the derivative of the natural logarithm at \( x = 1 \).
Natural Logarithm
A natural logarithm is a logarithm with the base \( e \), where \( e \approx 2.718 \). The function \( \ln(x) \) represents the natural logarithm of \( x \). It is particularly useful in various mathematical areas because it relates to exponential growth and decay.
  • Characteristics: The function \( \ln(x) \) is only defined for positive real numbers. It is an increasing function, meaning it always grows larger as \( x \) increases.
  • Derivative: As seen in the exercise, the derivative of \( \ln(x) \) is \( \frac{1}{x} \). This tells us the rate of change of \( \ln(x) \) at any point \( x \).
We used the property that \( \ln(1) = 0 \) to simplify our calculations in the exercise.
Limits
The concept of limits is fundamental in calculus. A limit helps define the value that a function approaches as the input gets closer to a certain point. In our problem, limits are crucial for defining derivatives and understanding the number \( e \).
  • Limit and Derivatives: Limits allow us to find derivatives by evaluating how a function behaves as it approaches a specific value. For instance, finding \( L'(1) \) required evaluating \[ \lim_{h \to 0} \frac{\ln(1 + h)}{h} \].
  • Calculating \( e \): The number \( e \) is defined using a specific limit. Knowing \[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} \] helps us in various exponential and calculus applications.
In our exercise, recognizing that \( e \) can be expressed this way allowed us to identify similar limit forms and calculate derivatives effectively.

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Most popular questions from this chapter

Let \(\left\\{r_{1}, r_{2}, \ldots, r_{\mathrm{n}} \ldots\right\\}\) be an enumeration of the rational numbers in \(I:=[0,1]\), and let \(f_{n}:\) \(I \rightarrow \mathbb{R}\) be defined to be 1 if \(x=r_{1}, \ldots, r_{n}\) and equal to 0 otherwise. Show that \(f_{n}\) is Riemann integrable for each \(n \in \mathbb{N}\), that \(f_{1}(x) \leq f_{2}(x) \leq \cdots \leq f_{n}(x) \leq \cdots\), and that \(f(x):=\lim \left(f_{n}(x)\right)\) is the Dirichlet function, which is not Riemann integrable on \([0,1]\).

Show that \(\lim \left(x^{2} e^{-n x}\right)=0\) and that \(\lim \left(n^{2} x^{2} e^{-n x}\right)=0\) for \(x \in \mathbb{R}, x \geq 0\).

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be uniformly continuous on \(\mathbb{R}\) and let \(f_{n}(x):=f(x+1 / n)\) for \(x \in \mathbb{R}\). Show that \(\left(f_{n}\right)\) converges uniformly on \(\mathbb{R}\) to \(f\).

Show that \(\lim \left((\cos \pi x)^{2 n}\right)\) exists for all \(x \in \mathbb{R} .\) What is its limit?

Evaluate \(\lim ((\sin n x) /(1+n x))\) for \(x \in \mathbb{R}, x \geq 0\).
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