Taylor Vs L'Hôpital's In Case Of X → ∞ X\to \infty X → ∞ Or ∞ ∞ \frac{\infty}{\infty} ∞ ∞ ​

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Introduction

When dealing with limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$, we often encounter the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. In such cases, two popular methods come to mind: L'Hôpital's theorem and Taylor's theorem. While L'Hôpital's rule is widely used and well-known, Taylor's theorem is often overlooked, despite its potential to provide a more elegant and insightful solution. In this article, we will explore the strengths and weaknesses of both methods, particularly in the case of $x\to \infty$ or $\frac{\infty}{\infty}$.

L'Hôpital's Theorem

L'Hôpital's theorem is a powerful tool for evaluating limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ when it takes the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The theorem states that if $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=0$ or $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=\infty$, then:

$$\lim\limits_{x\to a}\frac{f(x)}{g(x)}=\lim\limits_{x\to a}\frac{f'(x)}{g'(x)}$$

where $f'(x)$ and $g'(x)$ are the derivatives of $f(x)$ and $g(x)$, respectively.

Taylor's Theorem

Taylor's theorem, on the other hand, is a more general result that provides a power series expansion of a function around a point. The theorem states that if a function $f(x)$ is infinitely differentiable at a point $a$, then:

$$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots$$

This power series expansion can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ by substituting the power series expansions of $f(x)$ and $g(x)$ into the limit.

Comparison of L'Hôpital's and Taylor's Theorems

While both L'Hôpital's theorem and Taylor's theorem can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$, they have different strengths and weaknesses.

L'Hôpital's theorem is often easier to apply, as it only requires the calculation of the derivatives of $f(x)$ and $g(x)$. However, it can be more difficult to apply when the derivatives are not easily calculable, or when the limit takes the form $\frac{\infty}{\infty}$.

Taylor's theorem, on the other hand, provides a more general result that can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ by substituting the power series expansions of $f(x)$ and $g(x)$ into the limit. However, it can be more difficult to apply, as it requires the calculation of the power series expansions of $f(x)$ and $g(x)$.

Case of $x\to \infty$

When dealing with limits of the form $\lim\limits_{x\to \infty}\frac{f(x)}{g(x)}$, L'Hôpital's theorem can be particularly useful. By applying L'Hôpital's theorem, we can often simplify the limit to a form that is easier to evaluate.

For example, consider the limit:

$$\lim\limits_{x\to \infty}\frac{x^2}{e^x}$$

By applying L'Hôpital's theorem, we get:

$$\lim\limits_{x\to \infty}\frac{x^2}{e^x}=\lim\limits_{x\to \infty}\frac{2x}{e^x}$$

We can continue to apply L'Hôpital's theorem until we get a limit that is easier to evaluate.

Case of $\frac{\infty}{\infty}$

When dealing with limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ that take the form $\frac{\infty}{\infty}$, Taylor's theorem can be particularly useful. By substituting the power series expansions of $f(x)$ and $g(x)$ into the limit, we can often simplify the limit to a form that is easier to evaluate.

For example, consider the limit:

$$\lim\limits_{x\to 0}\frac{\sin x}{x}$$

By substituting the power series expansions of $\sin x$ and $x$ into the limit, we get:

$$\lim\limits_{x\to 0}\frac{\sin x}{x}=\lim\limits_{x\to 0}\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots}{x}$$

We can simplify the limit by canceling out the $x$ terms:

$$\lim\limits_{x\to 0}\frac{\sin x}{x}=\lim\limits_{x\to 0}\left(1-\frac{x^2}{3!}+\frac{x^4}{5!}-\cdots\right)$$

We can continue to simplify the limit by evaluating the remaining terms.

Conclusion

In conclusion, both L'Hôpital's theorem and Taylor's theorem can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$. While L'Hôpital's theorem is often easier to apply, Taylor's theorem provides a more general result that can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ by substituting the power series expansions of $f(x)$ and $g(x)$ into the limit. In the case of $x\to \infty$ or $\frac{\infty}{\infty}$, Taylor's theorem can be particularly useful, as it provides a more elegant and insightful solution.

References

• L'Hôpital, G. F. A. (1696). Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. Paris: Imprimerie Royale.
• Taylor, B. (1715). Methodus Incrementorum Directa et Inversa. London: J. Tonson.
• Spivak, M. (1965). Calculus. New York: W.A. Benjamin.

Appendix

The following is a list of common limits that can be evaluated using L'Hôpital's theorem and Taylor's theorem:

• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are rational functions
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are trigonometric functions
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are exponential functions

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Q: What is the main difference between L'Hôpital's theorem and Taylor's theorem?

A: The main difference between L'Hôpital's theorem and Taylor's theorem is that L'Hôpital's theorem is a specific result that can be used to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ when it takes the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Taylor's theorem, on the other hand, is a more general result that provides a power series expansion of a function around a point.

Q: When should I use L'Hôpital's theorem and when should I use Taylor's theorem?

A: You should use L'Hôpital's theorem when the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and the derivatives of $f(x)$ and $g(x)$ are easily calculable. You should use Taylor's theorem when the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and the power series expansions of $f(x)$ and $g(x)$ are easily calculable.

Q: Can I use both L'Hôpital's theorem and Taylor's theorem to evaluate a limit?

A: Yes, you can use both L'Hôpital's theorem and Taylor's theorem to evaluate a limit. However, you should be careful not to use both methods in a way that cancels out the work you have already done.

Q: How do I apply L'Hôpital's theorem to evaluate a limit?

A: To apply L'Hôpital's theorem, you need to follow these steps:

1. Check if the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
2. Calculate the derivatives of $f(x)$ and $g(x)$.
3. Substitute the derivatives into the limit.
4. Evaluate the resulting limit.

Q: How do I apply Taylor's theorem to evaluate a limit?

A: To apply Taylor's theorem, you need to follow these steps:

1. Check if the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
2. Calculate the power series expansions of $f(x)$ and $g(x)$.
3. Substitute the power series expansions into the limit.
4. Evaluate the resulting limit.

Q: What are some common limits that can be evaluated using L'Hôpital's theorem and Taylor's theorem?

A: Some common limits that can be evaluated using L'Hôpital's theorem and Taylor's theorem include:

• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are rational functions
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are trigonometric functions
• $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are exponential functions

Q: What are some tips for using L'Hôpital's theorem and Taylor's theorem?

A: Some tips for using L'Hôpital's theorem and Taylor's theorem include:

• Make sure to check if the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying either theorem.
• Calculate the derivatives or power series expansions carefully to avoid errors.
• Use both theorems in a way that cancels out the work you have already done.
• Be careful when evaluating the resulting limit to avoid making mistakes.

Q: What are some common mistakes to avoid when using L'Hôpital's theorem and Taylor's theorem?

A: Some common mistakes to avoid when using L'Hôpital's theorem and Taylor's theorem include:

• Not checking if the limit takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying either theorem.
• Calculating the derivatives or power series expansions incorrectly.
• Using both theorems in a way that cancels out the work you have already done.
• Making mistakes when evaluating the resulting limit.

Q: Can I use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to \infty}\frac{f(x)}{g(x)}$?

A: Yes, you can use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to \infty}\frac{f(x)}{g(x)}$. However, you should be careful when applying L'Hôpital's theorem, as it may not always be applicable.

Q: Can I use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not differentiable at $a$?

A: No, you cannot use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not differentiable at $a$. In this case, you may need to use other methods, such as the squeeze theorem or the definition of a limit.

Q: Can I use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not continuous at $a$?

A: No, you cannot use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not continuous at $a$. In this case, you may need to use other methods, such as the squeeze theorem or the definition of a limit.

Q: Can I use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not defined at $a$?

A: No, you cannot use L'Hôpital's theorem and Taylor's theorem to evaluate limits of the form $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are not defined at $a$. In this case, you may need to use other methods, such as the squeeze theorem or the definition of a limit.