Find the Limit of (x - sqrt(x)) as x approaches infinity

TL;DR

X minus square root of X limit as X approaches Infinity equals Infinity due to X growing faster.

Transcript

hi in this video we're going to find the limit as X approaches Infinity of x minus the square root of x so there's a couple ways to do this so one you can just use intuition if you think about it X grows much faster than the square root of x and so when X approaches Infinity this is getting much bigger than this so basically you're going to get inf... Read More

Key Insights

• ☺️ As X approaches Infinity, X grows much faster than the square root of X.
• 😑 Dividing by X and rationalizing the expression simplifies the limit calculation process.
• 😑 The difference of squares formula aids in simplifying the expression and understanding the result.
• ☺️ Justifying the limit as X approaches Infinity involves understanding the algebraic steps involved.
• ☺️ Utilizing different techniques like dividing by X helps in handling complex limit calculations.
• ☺️ The limit of X minus square root of X as X approaches Infinity approaches infinity due to X growing significantly faster.
• ⛔ Algebraic manipulations like distributing and simplifying help in clarifying the limit result.

Questions & Answers

Q: Why does X minus square root of X limit approach Infinity as X approaches Infinity?

X grows faster than the square root of X, leading to an infinite result as X approaches infinity. Algebraically, dividing by X helps in simplifying and justifying this limit.

Q: How does dividing by X and rationalizing the expression help in finding the limit?

Dividing by X and rationalizing transforms the limit expression into a more manageable form, making it easier to see that the limit approaches Infinity as X grows.

Q: What is the significance of using the difference of squares formula in finding the limit?

Utilizing the difference of squares formula helps in simplifying the expression and showcasing that as X approaches infinity, the limit is indefinite, leading to Infinity as the result.

Q: Why is it important to understand the algebraic steps in finding limits as X approaches Infinity?

Understanding the algebraic manipulation and reasoning behind limit calculations helps in justifying the result and provides clarity on why the limit behaves a certain way as X approaches Infinity.

Summary & Key Takeaways

• X grows faster than square root of X when approaching Infinity.

• Algebraically, dividing by X simplifies the limit to X - 1 + 1/square root of X.

• Understanding the algebraic steps helps justify why the limit approaches infinity.