How would you go about solving this? Limit of x/sqrt(x^2+1) as x goes to infinity. Reddit inf/inf

Summary

This educational video script explains how to solve the limit of a function as U approaches infinity. The function in question is U over the square root of U squared plus one. The presenter first identifies the indeterminate form 'infinity over infinity' and then simplifies the expression by dividing both the numerator and the denominator by U, the highest power of U present. The script also mentions an alternative quick method for solving such limits by comparing the dominant terms. The presenter concludes by suggesting that viewers try using L'Hôpital's rule for this problem and invites them to leave comments if they have any questions.

Takeaways

• 🔍 The problem involves finding the limit of a function as U approaches infinity.
• 🌌 The initial form of the limit is indeterminate (infinity over infinity).
• 📚 To solve it, the script suggests dividing both the numerator and the denominator by the highest power of U, which is \( U^2 \) in this case.
• 🔢 The script demonstrates the process by rewriting the square root of \( U^2 \) as \( U \) and then dividing both the top and bottom by \( U \).
• 📉 After simplification, the limit is found to be \( 1 \) as \( U \) approaches infinity.
• 🚀 The script also mentions an alternative quick method for solving such limits by comparing the highest powers of the variable in the numerator and the denominator.
• 📌 It's important to note that the sign of the limit (positive or negative infinity) depends on the direction from which U approaches infinity.
• 🤔 The script hints at a potential issue when applying L'Hôpital's rule to this problem, suggesting it might yield unexpected results.
• 💬 The speaker encourages viewers to ask questions in the comments if they have any.
• 🔄 The process of solving the limit is shown in a step-by-step manner, making it easy to follow.

Q & A

What is the initial indeterminate form of the limit problem presented in the script?

The initial indeterminate form is 'infinity over infinity', which arises when evaluating the limit as U approaches infinity of U over the square root of U^2 + 1.

Why is the square root of U^2 + 1 simplified to U as U approaches infinity?

The square root of U^2 + 1 is simplified to U because the dominant term in the expression under the square root is U^2, and as U approaches infinity, the term +1 becomes negligible compared to U^2.

How does dividing the numerator and the denominator by U to the first power help in solving the limit?

Dividing both the numerator and the denominator by U to the first power simplifies the expression, allowing us to cancel out the U terms and leaving us with a limit that can be evaluated more easily.

What is the result of the limit after applying the simplification method described in the script?

After simplification, the limit evaluates to 1, as the dominant term in the numerator (U) divided by the dominant term in the denominator (U) yields 1, and the remaining terms approach zero as U approaches infinity.

What is the alternative quick method mentioned for evaluating limits involving infinity?

The alternative quick method involves comparing the highest power of the variable in both the numerator and the denominator, effectively canceling out those terms and leaving a simpler expression to evaluate.

Why is it important to consider the sign of U when applying the quick method for evaluating limits?

The sign of U is important because it determines the direction from which infinity is approached. If U approaches negative infinity, the result of the limit could be negative, which is not considered in the standard application of the method.

What happens if you attempt to use L'Hôpital's Rule on this limit problem?

Attempting to use L'Hôpital's Rule on this problem might lead to a 'zero over zero' indeterminate form, which would require further differentiation, as the direct application of the rule would not yield a conclusive result.

Why does the script suggest trying L'Hôpital's Rule and observing what happens?

The script suggests trying L'Hôpital's Rule as an exercise to understand the limitations and peculiarities of the rule, especially when dealing with limits that initially result in indeterminate forms.

What is the significance of the term 'indeterminate form' in calculus?

The term 'indeterminate form' refers to a situation where the limit of a function does not exist or cannot be determined from the direct substitution of the variable, often resulting in expressions like '0/0' or '∞/∞'.

How does the script demonstrate the process of evaluating limits involving infinity?

The script demonstrates the process by first identifying the indeterminate form, then applying algebraic manipulations to simplify the expression, and finally evaluating the limit by considering the dominant terms as the variable approaches infinity.