Problem 15 - Basic Analysis: Introduction to Real Analysis

Question

Prove the following simple version of \(L\) 'Hôpital's rule. Suppose \(f:(a, b) \rightarrow \mathbb{R}\) and \(g:(a, b) \rightarrow \mathbb{R}\) are differentiable functions whose derivatives \(f^{\prime}\) and \(g^{\prime}\) are continuous functions. Suppose that at \(c \in(a, b), f(c)=0, g(c)=0,\) and \(g^{\prime}(x) \neq 0\) for all \(x \in(a, b),\) and suppose that the limit of \(f^{\prime}(x) / g^{\prime}(x)\) as \(x\) goes to \(c\) exists. Show that $$\lim _{x \rightarrow c} \frac{f(x)}{g(x)}=\lim _{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)}.$$

Step-by-Step Solution

Answer

\(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)}\) using Cauchy's Mean Value Theorem.

Step 1: Understand the Conditions

We are given functions \(f\) and \(g\) that are differentiable on \((a, b)\), with \(f(c) = 0\) and \(g(c) = 0\), for some point \(c \in (a, b)\). Additionally, \(g^{\prime}(x) eq 0\) for all \(x \in (a, b)\), and the limit of \(\frac{f^{\prime}(x)}{g^{\prime}(x)}\) as \(x\) approaches \(c\) exists. We need to show that \(\lim _{x \rightarrow c} \frac{f(x)}{g(x)} = \lim _{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)}\)."

Step 2: Apply Cauchy's Mean Value Theorem

According to Cauchy's Mean Value Theorem, for two functions \(f\) and \(g\) that are continuous and differentiable on \([c, x]\), there exists some \(\xi\) in \((c, x)\) such that \(\frac{f(x)-f(c)}{g(x)-g(c)} = \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}\). Since \(f(c) = 0\) and \(g(c) = 0\), this simplifies to \(\frac{f(x)}{g(x)} = \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}\)."

Step 3: Take the Limit as \(x\) Approaches \(c\)

Taking the limit of both sides as \(x\) approaches \(c\), we get \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}\). By hypothesis, we know that \(\lim_{x \to c} \frac{f^{\prime}(x)}{g^{\prime}(x)}\) exists. Since \(\xi\) converges to \(c\) as \(x\) does, we have \(\lim_{\xi \to c} \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)} = \lim_{x \to c} \frac{f^{\prime}(x)}{g^{\prime}(x)}\), thereby proving the result."

Key Concepts

Cauchy's Mean Value Theorem

Cauchy's Mean Value Theorem is an extension of the Mean Value Theorem in calculus. It plays a crucial role in the proof of L'Hôpital's Rule, especially in the context of differentiable functions.
The theorem states that for two functions, say \( f \) and \( g \), that are continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \( \xi \) in \((a, b)\) such that:

• \( \frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(ib)}{g'(ib)} \)

This form is particularly useful when you have two functions and you want to relate their increments through their derivatives.
When you apply Cauchy's Mean Value Theorem to the functions mentioned in our problem, you essentially get the version of the formula that assists you in linking \( \frac{f(x)-f(a)}{g(x)-g(a)} \) and \( \frac{f'(\xi)}{g'(\xi)} \), helping you move further in the steps to prove L'Hôpital's Rule. This connection is vital because it allows you to move from a statement about the functions to a statement about their derivatives.

Differentiable Functions

Differentiable functions are fundamental in calculus because they allow us to study how functions change at every point in their domain. A function \( f \) is said to be differentiable at a point \( c \) if the derivative \( f'(c) \) exists, meaning you can calculate a limit that defines the rate of change of the function at that specific point.
Differentiable functions have certain properties that make them ideal candidates for applying calculus theorems, such as L'Hôpital's Rule. They must be smooth, with no sharp corners or discontinuities at the points of interest.
For a function to be differentiable over an entire interval, it must be smooth across that interval. This smoothness is why you can apply a theorem like Cauchy's Mean Value Theorem, which requires functions to be both continuous and differentiable. Therefore, differentiability is a necessary check before applying related theorems or techniques, ensuring you legally proceed with calculations and transformations critical in proving calculus principles.

Limit Calculation

Limit calculation is a core part of calculus and is vital when working with L'Hôpital's Rule. Limiting processes are used to find the tendency of a function as the input approaches a certain value. In the given problem, the understanding of limits allows us to prove the behavior of \( \frac{f(x)}{g(x)} \) as \( x \) approaches \( c \).
L'Hôpital's Rule specifically deals with indeterminate forms like \( \frac{0}{0} \), which requires calculating limits to unravel the undefined nature at the point \( c \).
To calculate these limits, one commonly uses algebraic manipulations, theorems like Cauchy's Mean Value Theorem, or other differentiation techniques. The goal is to find a more easily evaluated expression.
Moreover, calculating limits often involves checking the continuity and differentiability of functions in the region of interest. By achieving a defined result via limits, we can affirmatively conclude the desired results, such as the limit equality in our problem, deducing the behavior of the function ratio through the behavior of their derivatives.