Find The Limit. Use L'Hôpital's Rule Where Appropriate. If There Is A More Elementary Method, Consider Using It. Lim ⁡ X → Π / 4 Cos ⁡ ( X ) − Sin ⁡ ( X ) Tan ⁡ ( X ) − 1 \lim _{x \rightarrow \pi / 4} \frac{\cos (x)-\sin (x)}{\tan (x)-1} Lim X → Π /4 ​ T A N ( X ) − 1 C O S ( X ) − S I N ( X ) ​

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Introduction

In mathematics, finding limits is a crucial concept that helps us understand the behavior of functions as the input values approach a specific point. In this article, we will explore the limit of a trigonometric function using various methods, including l'Hôpital's Rule. We will also discuss the importance of considering more elementary methods before resorting to advanced techniques.

The Problem

The given problem is to find the limit of the function cos⁡(x)−sin⁡(x)tan⁡(x)−1\frac{\cos (x)-\sin (x)}{\tan (x)-1} as xx approaches π4\frac{\pi}{4}. This is a classic example of a limit problem that requires careful analysis and application of mathematical techniques.

Elementary Method: Direct Substitution

One of the first methods to try when finding a limit is direct substitution. This involves substituting the value of xx into the function and evaluating the result. In this case, we can substitute x=π4x = \frac{\pi}{4} into the function:

cos⁡(π4)−sin⁡(π4)tan⁡(π4)−1\frac{\cos \left(\frac{\pi}{4}\right)-\sin \left(\frac{\pi}{4}\right)}{\tan \left(\frac{\pi}{4}\right)-1}

Using the values of the trigonometric functions at π4\frac{\pi}{4}, we get:

12−121−1\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{1-1}

Unfortunately, this results in an indeterminate form of 00\frac{0}{0}, which means that direct substitution is not a viable method in this case.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful technique for finding limits of functions that result in indeterminate forms. The rule states that if a limit is of the form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, we can differentiate the numerator and denominator separately and then take the limit of the resulting quotient.

In this case, we can apply L'Hôpital's Rule by differentiating the numerator and denominator:

lim⁡x→π4cos⁡(x)−sin⁡(x)tan⁡(x)−1=lim⁡x→π4−sin⁡(x)−cos⁡(x)sec⁡2(x)\lim_{x \to \frac{\pi}{4}} \frac{\cos (x)-\sin (x)}{\tan (x)-1} = \lim_{x \to \frac{\pi}{4}} \frac{-\sin (x)-\cos (x)}{\sec^2 (x)}

Now, we can substitute x=π4x = \frac{\pi}{4} into the resulting expression:

lim⁡x→π4−sin⁡(x)−cos⁡(x)sec⁡2(x)=−sin⁡(π4)−cos⁡(π4)sec⁡2(π4)\lim_{x \to \frac{\pi}{4}} \frac{-\sin (x)-\cos (x)}{\sec^2 (x)} = \frac{-\sin \left(\frac{\pi}{4}\right)-\cos \left(\frac{\pi}{4}\right)}{\sec^2 \left(\frac{\pi}{4}\right)}

Using the values of the trigonometric functions at π4\frac{\pi}{4}, we get:

−12−122\frac{-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{2}

Simplifying the expression, we get:

−22\frac{-\sqrt{2}}{2}

Conclusion

In this article, we have explored the limit of a trigonometric function using various methods, including direct substitution and L'Hôpital's Rule. We have seen that direct substitution is not a viable method in this case, but L'Hôpital's Rule provides a powerful technique for finding the limit. The resulting limit is −22\frac{-\sqrt{2}}{2}.

Final Answer

The final answer is −22\boxed{\frac{-\sqrt{2}}{2}}.

Additional Resources

For more information on limits and L'Hôpital's Rule, please refer to the following resources:

Related Problems

Tags

  • Limits
  • L'Hôpital's Rule
  • Trigonometric functions
  • Calculus
  • Mathematics

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Introduction

In our previous article, we explored the limit of a trigonometric function using various methods, including direct substitution and L'Hôpital's Rule. In this article, we will answer some common questions related to finding limits with L'Hôpital's Rule.

Q: What is L'Hôpital's Rule?

A: L'Hôpital's Rule is a mathematical technique used to find the limit of a function that results in an indeterminate form, such as 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}. The rule states that we can differentiate the numerator and denominator separately and then take the limit of the resulting quotient.

Q: When can I use L'Hôpital's Rule?

A: You can use L'Hôpital's Rule when the limit of a function results in an indeterminate form, such as 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}. However, you should only use L'Hôpital's Rule when the function is differentiable at the point where the limit is being evaluated.

Q: How do I apply L'Hôpital's Rule?

A: To apply L'Hôpital's Rule, you need to differentiate the numerator and denominator separately and then take the limit of the resulting quotient. You can use the power rule, product rule, and quotient rule to differentiate the functions.

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

A: Some common mistakes to avoid when using L'Hôpital's Rule include:

  • Not checking if the function is differentiable at the point where the limit is being evaluated
  • Not differentiating the numerator and denominator correctly
  • Not taking the limit of the resulting quotient
  • Using L'Hôpital's Rule when the function is not in an indeterminate form

Q: Can I use L'Hôpital's Rule to find the limit of a function that is not in an indeterminate form?

A: No, you should not use L'Hôpital's Rule to find the limit of a function that is not in an indeterminate form. L'Hôpital's Rule is only applicable when the function is in an indeterminate form, such as 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}.

Q: What are some common applications of L'Hôpital's Rule?

A: L'Hôpital's Rule has many applications in mathematics, including:

  • Finding the limit of a function that results in an indeterminate form
  • Evaluating the limit of a function at a point where the function is not defined
  • Finding the derivative of a function using the definition of a derivative
  • Solving optimization problems

Q: Can I use L'Hôpital's Rule to find the limit of a function that involves trigonometric functions?

A: Yes, you can use L'Hôpital's Rule to find the limit of a function that involves trigonometric functions. In fact, L'Hôpital's Rule is often used to find the limit of functions that involve trigonometric functions, such as sine, cosine, and tangent.

Q: What are some common limits that involve trigonometric functions?

A: Some common limits that involve trigonometric functions include:

  • lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}
  • lim⁡x→0tan⁡xx\lim_{x \to 0} \frac{\tan x}{x}
  • lim⁡x→π4cos⁡x−sin⁡xtan⁡x−1\lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\tan x - 1}

Conclusion

In this article, we have answered some common questions related to finding limits with L'Hôpital's Rule. We have discussed the definition of L'Hôpital's Rule, when to use it, how to apply it, and some common mistakes to avoid. We have also discussed some common applications of L'Hôpital's Rule and some common limits that involve trigonometric functions.

Final Answer

The final answer is −22\boxed{\frac{-\sqrt{2}}{2}}.

Additional Resources

For more information on L'Hôpital's Rule and limits, please refer to the following resources:

Related Problems

Tags

  • Limits
  • L'Hôpital's Rule
  • Trigonometric functions
  • Calculus
  • Mathematics