Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

L'Hôpital's Rule in its 1696 form said this: If \(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} g(x)=0\), then \(\lim _{x \rightarrow a} f(x) / g(x)=f^{\prime}(a) / g^{\prime}(a)\), provided that \(f^{\prime}(a)\) and \(g^{\prime}(a)\) both exist and \(g^{\prime}(a) \neq 0\). Prove this result without recourse to Cauchy's Mean Value Theorem.

Step-by-Step Solution

Answer

\(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}\).

Step 1: Definition and Conditions

We have two functions, \(f(x)\) and \(g(x)\), both of which approach zero as \(x\) approaches \(a\): \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\). The derivatives \(f'(a)\) and \(g'(a)\) exist, and \(g'(a) eq 0\). We aim to find \(\lim_{x \to a} \frac{f(x)}{g(x)}\).

Step 2: Apply Taylor Approximation

We can approximate the functions \(f(x)\) and \(g(x)\) near \(x = a\) by their Taylor expansions: - \(f(x) = f(a) + f'(a)(x-a) + \text{higher order terms}\) - \(g(x) = g(a) + g'(a)(x-a) + \text{higher order terms}\)Since \(f(a) = 0\) and \(g(a) = 0\), the expressions simplify to:\(f(x) \approx f'(a)(x-a)\) and \(g(x) \approx g'(a)(x-a)\).

Step 3: Form the Quotient

We form the quotient of the approximations:\(\frac{f(x)}{g(x)} \approx \frac{f'(a)(x-a)}{g'(a)(x-a)}\).Here, the \((x - a)\) terms cancel out, so the expression simplifies to:\(\frac{f'(a)}{g'(a)}\).

Step 4: Consider the Limit

As \(x\) approaches \(a\), the approximation becomes more precise. Therefore:\(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(a)}{g'(a)}\).Since the \((x-a)\) terms cancel out immediately as \(x\) approaches \(a\), we directly find:\(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}\).

Step 5: Conclusion

We conclude that L'Hôpital's Rule in its 1696 form is proven without using Cauchy's Mean Value Theorem under the given conditions: If \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\), then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}\), provided that \(f'(a)\) and \(g'(a)\) exist, and \(g'(a) eq 0\).

Key Concepts

Taylor Approximation

Taylor Approximation is a powerful mathematical tool that lets us express functions as sums of their derivatives. When we say we "approximate" a function using Taylor series, we mean we are writing it out as a series of terms that include its derivatives at a specific point.

The basic idea is to expand a function around a point, often denoted as "a". Imagine you have a function, and you want to know its behavior as it gets close to some point. Taylor Approximation helps by simplifying the function using derivatives.

• The linear approximation of a function at a point "a" is given by: \[ f(x) \approx f(a) + f'(a)(x-a) \] This is referred to as the Taylor expansion of the function at that point.
• For higher accuracy, more terms can be added: \[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \] This allows us to precisely describe the function's behavior as it approaches "a".

In the context of L'Hôpital's Rule, the Taylor approximation is key because it lets us see how two functions behave around a point where both go to zero.

Limit of a Function

The limit of a function is a fundamental concept in calculus, helping us understand the value that a function approaches as the input reaches a certain point. Basically, limits describe what happens to a function as the variable within it gets infinitely close to a particular value.

When we write \(\lim_{x \to a} f(x)\), we are saying "as \(x\) gets closer and closer to \(a\), what value does \(f(x)\) approach?" If \(f(x)\) becomes closer to some number, that number is the limit of \(f(x)\) as \(x\) approaches \(a\).

• Limits are crucial for evaluating indeterminate forms, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), where the values are not clear just through direct substitution.
• L'Hôpital's Rule specifically utilizes the limit when both functions \(f(x)\) and \(g(x)\) go to zero at the same point, allowing us to switch focus from \(f(x)\) and \(g(x)\) to their derivatives.

Understanding limits helps in proving L'Hôpital's Rule because it involves substituting the original limit of two functions for the limit of their derivatives.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes as its input changes. It's like asking, "How fast is the car going right now?" when looking at a car's speed function. The derivative can be thought of as a slope or a rate of change.
- Given a function \(f(x)\), the derivative \(f'(x)\) is formulated as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] This formula captures how much \(f(x)\) changes as \(x\) increases by a tiny amount "h".- The derivative tells us lots about the function: - Increasing or decreasing behavior: If \(f'(x) > 0\), the function is increasing at \(x\), while if \(f'(x) < 0\), it's decreasing. - Points of extrema: Where \(f'(x) = 0\), the function might have a local maximum or minimum.In the context of L'Hôpital's Rule, derivatives are the core elements that allow us to solve indeterminate forms like \(\frac{0}{0}\) by replacing the original functions with their derivatives. This technique works because the derivatives provide much clearer insight into the behavior of the functions near the point of interest.