What Is the Limit of x*tan^-1(x) as x Approaches Infinity?

TL;DR

The limit of x*tan^-1(x) as x approaches infinity is infinity. Although L'Hopital's Rule is often a go-to method for limits in indeterminate forms, it cannot be applied here due to the nature of the expression. Instead, recognizing that tan^-1(x) approaches π/2 helps conclude that the limit results in infinity.

Transcript

okay let's calculate the limit as X goes  to infinity of x*tan^-1(x)   so it seems like we have a product two  things I should bring the X down down right   in other words I should look at this as the  limit as X goes to infinity let me keep the   inverse tangent X on the top and once I bring  the X down down it becomes one all for X and   the rea... Read More

Key Insights

• ☺️ The video demonstrates the use of L'Hopital's Rule to calculate the limit of x*tan^-1(x) as x approaches infinity.
• 💼 However, it is revealed that L'Hopital's Rule cannot be used in this specific case.
• ⛔ The inverse tangent function behavior is utilized to determine the actual value of the limit.
• 😨 Care should be taken when applying L'Hopital's Rule, ensuring the limit is in the appropriate form.
• ♾️ The limit of x*tan^-1(x) as x approaches infinity is infinity, contrary to the initial expectation of a finite result.
• 😑 The specific nature of the expression, x*tan^-1(x), prevents the application of L'Hopital's Rule.
• 💁 The video highlights the importance of identifying the appropriate limit form for applying L'Hopital's Rule.

Questions & Answers

Q: How can we calculate the limit of x*tan^-1(x) as x approaches infinity?

To calculate this limit, we initially consider applying L'Hopital's Rule. By taking the derivative of the expression and simplifying, the limit can be found. However, we discover that L'Hopital's Rule is not applicable in this case.

Q: Why is L'Hopital's Rule not applicable in this limit calculation?

L'Hopital's Rule requires the limit to be in the form of 0/0 or infinity/infinity. In the expression x*tan^-1(x), plugging in infinity does not result in 0/0 or infinity/infinity, making L'Hopital's Rule inappropriate.

Q: What is the value of the limit of x*tan^-1(x) as x approaches infinity?

The video explains that since L'Hopital's Rule cannot be used, an alternative approach is needed. By considering the behavior of the inverse tangent function, it is determined that the limit of x*tan^-1(x) as x approaches infinity is infinity.

Q: What should be taken into consideration when solving limit questions using L'Hopital's Rule?

It is essential to ensure that the limit presents a 0/0 or infinity/infinity scenario for L'Hopital's Rule to be applicable. Without this specific form, applying L'Hopital's Rule would lead to an incorrect solution.

Summary & Key Takeaways

• The video discusses calculating the limit of x*tan^-1(x) as x approaches infinity.

• L'Hopital's Rule is proposed as a method to solve the limit by taking the derivative of the expression.

• However, it is revealed that L'Hopital's Rule cannot be applied in this case due to the specific nature of the expression.