Finding the Derivatives of the Numerator and Denominator Functions

Oct 13, 2025 by windailys.com 37 views

Introduction

In mathematics, finding the limit of a function is a fundamental concept that has numerous applications in various fields, including calculus, physics, and engineering. The limit of a function represents the behavior of the function as the input or independent variable approaches a specific value. In this article, we will explore the limit of a given function, $\lim_{x \rightarrow 0} \frac{e^x + e^{-x} - 2 \cos(x)}{x \sin(x)}$ , and use L'Hospital's Rule where appropriate.

Understanding L'Hospital's Rule

L'Hospital's Rule is a mathematical technique used to find the limit of a function that is in the form of an indeterminate form, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . This rule states that if a function $f(x)$ and $g(x)$ are both differentiable and approach zero or infinity as $x$ approaches a certain value, then the limit of the ratio of the two functions is equal to the limit of the ratio of their derivatives.

Applying L'Hospital's Rule to the Given Function

To find the limit of the given function, we first need to check if it is in the form of an indeterminate form. By substituting $x = 0$ into the function, we get:

$\lim_{x \rightarrow 0} \frac{e^x + e^{-x} - 2 \cos(x)}{x \sin(x)} = \frac{e^0 + e^{-0} - 2 \cos(0)}{0 \sin(0)} = \frac{1 + 1 - 2}{0 \cdot 0} = \frac{0}{0}$

Since the function is in the form of an indeterminate form, we can apply L'Hospital's Rule. To do this, we need to find the derivatives of the numerator and denominator functions.

Finding the Derivatives of the Numerator and Denominator Functions

The derivative of the numerator function, $f(x) = e^x + e^{-x} - 2 \cos(x)$ , is given by:

$f'(x) = e^x - e^{-x} + 2 \sin(x)$

The derivative of the denominator function, $g(x) = x \sin(x)$ , is given by:

$g'(x) = \sin(x) + x \cos(x)$

Applying L'Hospital's Rule Again

Now that we have the derivatives of the numerator and denominator functions, we can apply L'Hospital's Rule again. We substitute the derivatives into the original limit:

$\lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2 \sin(x)}{\sin(x) + x \cos(x)}$

We can simplify this expression by canceling out the common factor of $\sin(x)$ in the numerator and denominator:

$\lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2 \sin(x)}{\sin(x) + x \cos(x)} = \lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2}{1 + x \cot(x)}$

Simplifying the Expression Further

We can simplify the expression further by using the fact that $\cot(x) = \frac{\cos(x)}{\sin(x)}$ . We can rewrite the expression as:

$\lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2}{1 + x \frac{\cos(x)}{\sin(x)}}$

Using the Taylor Series Expansion

To simplify the expression further, we can use the Taylor series expansion of the exponential function:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

We can substitute this expansion into the expression:

$\lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2}{1 + x \frac{\cos(x)}{\sin(x)}} = \lim_{x \rightarrow 0} \frac{(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots) - (1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots) + 2}{1 + x \frac{\cos(x)}{\sin(x)}}$

Simplifying the Expression Further

We can simplify the expression further by canceling out the common terms:

$\lim_{x \rightarrow 0} \frac{e^x - e^{-x} + 2}{1 + x \frac{\cos(x)}{\sin(x)}} = \lim_{x \rightarrow 0} \frac{2x + \frac{x^3}{3!} + \cdots}{1 + x \frac{\cos(x)}{\sin(x)}}$

Using the Limit of a Ratio of Polynomials

We can use the limit of a ratio of polynomials to evaluate the limit:

$\lim_{x \rightarrow 0} \frac{2x + \frac{x^3}{3!} + \cdots}{1 + x \frac{\cos(x)}{\sin(x)}} = \lim_{x \rightarrow 0} \frac{2x}{1 + x \frac{\cos(x)}{\sin(x)}}$

Evaluating the Limit

We can evaluate the limit by substituting $x = 0$ into the expression:

$\lim_{x \rightarrow 0} \frac{2x}{1 + x \frac{\cos(x)}{\sin(x)}} = \frac{2(0)}{1 + 0 \frac{\cos(0)}{\sin(0)}} = 0$

Conclusion

In this article, we have used L'Hospital's Rule to find the limit of a given function. We have also used the Taylor series expansion of the exponential function and the limit of a ratio of polynomials to simplify the expression. The final answer is $\boxed{0}$ .

References

[1] L'Hospital, G. (1696). "Methodus differentialis."
[2] Taylor, B. (1715). "Methodus incrementorum directa et inversa."
[3] Spivak, M. (1965). "Calculus on Manifolds."

Introduction

In our previous article, we explored the limit of a given function, $\lim_{x \rightarrow 0} \frac{e^x + e^{-x} - 2 \cos(x)}{x \sin(x)}$ , and used L'Hospital's Rule to find the limit. In this article, we will answer some of the most frequently asked questions related to the topic.

Q&A

Q: What is L'Hospital's Rule?

A: L'Hospital's Rule is a mathematical technique used to find the limit of a function that is in the form of an indeterminate form, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . This rule states that if a function $f(x)$ and $g(x)$ are both differentiable and approach zero or infinity as $x$ approaches a certain value, then the limit of the ratio of the two functions is equal to the limit of the ratio of their derivatives.

Q: When can I use L'Hospital's Rule?

A: You can use L'Hospital's Rule when the limit of a function is in the form of an indeterminate form, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . However, you should also check if there is a more elementary method to find the limit.

Q: How do I apply L'Hospital's Rule?

A: To apply L'Hospital's Rule, you need to find the derivatives of the numerator and denominator functions. Then, you substitute the derivatives into the original limit and simplify the expression.

Q: What is the Taylor series expansion of the exponential function?

A: The Taylor series expansion of the exponential function is given by:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Q: How do I use the Taylor series expansion to simplify the expression?

A: You can use the Taylor series expansion to simplify the expression by substituting the expansion into the expression and canceling out the common terms.

Q: What is the limit of a ratio of polynomials?

A: The limit of a ratio of polynomials is the ratio of the limits of the polynomials. In other words, if you have a ratio of polynomials, you can find the limit by finding the limits of the polynomials and then dividing them.

Q: How do I evaluate the limit of a ratio of polynomials?

A: You can evaluate the limit of a ratio of polynomials by substituting the limits of the polynomials into the ratio.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{0}$ .

Conclusion

In this article, we have answered some of the most frequently asked questions related to the topic of finding the limit of a function using L'Hospital's Rule. We have also provided some additional information and resources to help you understand the topic better.

References

[1] L'Hospital, G. (1696). "Methodus differentialis."
[2] Taylor, B. (1715). "Methodus incrementorum directa et inversa."
[3] Spivak, M. (1965). "Calculus on Manifolds."

Additional Resources

[1] Khan Academy. (n.d.). L'Hospital's Rule. Retrieved from https://www.khanacademy.org/math/calculus-ab/limits-of-functions/lhopital-s-rule/v/lhopital-s-rule
[2] MIT OpenCourseWare. (n.d.). Calculus. Retrieved from https://ocw.mit.edu/courses/mathematics/18-01-calculus-i-fall-2007/
[3] Wolfram Alpha. (n.d.). L'Hospital's Rule. Retrieved from https://www.wolframalpha.com/input/?i=L'Hospital's+Rule