Problem 85 - Calculus Early Transcendentals

Question

Suppose \(f\) and \(g\) have Taylor series about the point \(a.\) a. If \(f(a)=g(a)=0\) and \(g^{\prime}(a) \neq 0,\) evaluate \(\lim _{x \rightarrow a} f(x) / g(x).\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with l'Hôpital's Rule. b. If \(f(a)=g(a)=f^{\prime}(a)=g^{\prime}(a)=0\) and \(g^{\prime \prime}(a) \neq 0\) evaluate \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with two applications of I'Hôpital's Rule.

Step-by-Step Solution

Answer

Answer: The limit of the ratio of \(f(x)\) to \(g(x)\) when \(x\) approaches \(a\) is \(\frac{f''(a)}{g''(a)}\).

Step 1: Write the Taylor series for f and g

We will first write the Taylor series for the functions \(f\) and \(g\) about the point \(a\). They can be written as: $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n $$ and $$ g(x) = \sum_{n=0}^\infty \frac{g^{(n)}(a)}{n!}(x-a)^n $$ Where \(f^{(n)}(a)\) and \(g^{(n)}(a)\) are the \(n\)-th order derivatives of the functions evaluated at \(a\).

Step 2: Apply the given conditions

We are given that \(f(a) = g(a) = 0\). Since the terms with \(n=0\) in Taylor series represent the value of functions at \(a\), we can remove these terms from both series: $$ f(x) = \sum_{n=1}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n $$ and $$ g(x) = \sum_{n=1}^\infty \frac{g^{(n)}(a)}{n!}(x-a)^n $$

Step 3: Evaluating the limit for part (a)

Our goal is to evaluate the limit as \(x \to a\) of their ratio: $$ \lim_{x \rightarrow a}\frac{f(x)}{g(x)} $$ Since both series start from \(n=1\), both functions have the same power of \((x-a)\) in their first term. Hence, we can evaluate this limit by looking at the first non-zero terms of both series and take the ratio of their coefficients: $$ \lim_{x \rightarrow a}\frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)} $$ This result is consistent with l'Hôpital's Rule when both functions have a limit of \(0\) as \(x\to a\). By applying l'Hôpital's Rule, we get: $$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} $$

Step 4: Evaluating the limit for part (b)

Now, for part (b), we are given that \(f'(a) = g'(a) = 0\) and \(g''(a) \neq 0\). The Taylor series for \(f\) and \(g\) starting from the first non-zero terms become: $$ f(x) = \sum_{n=2}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n $$ and $$ g(x) = \sum_{n=2}^\infty \frac{g^{(n)}(a)}{n!}(x-a)^n $$ Again, our goal is to evaluate the limit as \(x \to a\) of their ratio: $$ \lim_{x \rightarrow a}\frac{f(x)}{g(x)} $$ Just like in Step 3, we can evaluate this limit by taking the ratio of their first non-zero term's coefficients: $$ \lim_{x \rightarrow a}\frac{f(x)}{g(x)} = \frac{f''(a)}{g''(a)} $$ This result is consistent with two applications of l'Hôpital's Rule when both functions have a limit of \(0\) as \(x\to a\) for both value and first derivative. By applying l'Hôpital's Rule twice, we get: $$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)} $$

Key Concepts

l'Hôpital's Rule

l'Hôpital's Rule is a powerful tool in calculus used to evaluate limits that result in indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter such a limit, l'Hôpital's Rule allows you to differentiate the numerator and denominator separately until you can evaluate the limit directly. This method can simplify complicated expressions and make solving for limits easier.

For example, consider functions \(f(x)\) and \(g(x)\) whose limits as \(x\rightarrow a\) yield an indeterminate form. By applying l'Hôpital's Rule, you can find:

• If \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \), then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\), assuming \(\lim_{x \to a} \frac{f'(x)}{g'(x)}\) exists.
• In some cases, applying l'Hôpital's Rule more than once might be necessary if the first attempt still results in an indeterminate form.

Utilizing l'Hôpital's Rule effectively can simplify and provide insight into limits that initially seem unsolvable. It is important to recognize when a situation is suitable for applying this rule by identifying the indeterminate forms correctly.

limits

In calculus, limits help us understand the behavior of functions as they approach certain points. A limit gives us insight into what value a function approaches as the input, \(x\), gets closer and closer to a particular point, \(a\). Understanding limits is fundamental to studying continuous functions and calculating derivatives.

To find a limit of the function \(f(x)\) as \(x\) approaches \(a\), denoted by \(\lim_{x \rightarrow a}f(x)\), we consider:

• Values closer and closer to \(a\) from both directions (left and right).
• The value \(f(x)\) approaches or would hypothetically reach at \(x=a\), even if it doesn't actually reach it.

Limits can address the behavior at points of discontinuity or when functions are undefined. They lay the foundation for defining derivatives and integrals. More importantly, knowing how to manipulate and evaluate limits allows us to explore calculus concepts like continuity, convergence, and the rate of change.

derivatives

Derivatives provide insight into how a function changes at any given point. The derivative of a function \(f(x)\) represents its rate of change with respect to \(x\). In terms of geometry, the derivative at a point gives the slope of the tangent line to the function's graph at that point.

Some key points about derivatives include:

• The derivative of \(f(x)\), denoted \(f'(x)\) or \(\frac{df}{dx}\), gives the best linear approximation of \(f\) near any chosen \(x\).
• Derivatives are calculated using the definition: \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\).
• They help solve real-world problems involving changing circumstances, like velocity, acceleration, and optimization.

To apply derivatives effectively, it's important to understand rules like the power rule, product rule, quotient rule, and chain rule. Mastery of these concepts ensures accurate and efficient calculations, and allows us to explore further into calculus, such as studying Taylor series and rates of change across various contexts.