Prove that sin(ln(x))/x approaches 0 as x approaches infinity

TL;DR

The video explains the proof that as x approaches infinity, the limit of sine(ln(x)/x) is equal to zero.

Transcript

hi everyone in this problem we're going to prove that the limit as x approaches infinity of the sine of the natural log of x over x equals zero so before we do the proof let me just briefly recall what this actually means mathematically so when you write the limit as x approaches infinity of f of x equals l where l is a real number this is the same... Read More

Key Insights

• ☺️ The limit as x approaches infinity can be proven by finding an appropriate positive number that satisfies the desired conditions.
• 😑 The use of scratch work allows for experimentation and simplification of the expression before officially presenting the proof.
• 😑 Properties of absolute values and boundedness of functions can be used to simplify the mathematical expressions.

Questions & Answers

Q: What does it mean when the limit of a function approaches a real number?

When the limit of a function approaches a real number, it means that as x approaches infinity, the function gets arbitrarily close to that real number.

Q: How does the presenter use scratch work to simplify the expression?

The scratch work allows the presenter to manipulate the expression and apply properties of absolute values and the sine function to simplify it step-by-step before starting the official proof.

Q: What is the significance of the Archimedean principle in the proof?

The Archimedean principle states that for any real number, there exists a larger number. It helps in finding a positive number m that satisfies the condition necessary for the proof.

Q: Why is it important to reinforce the steps taken in the scratch work during the proof?

Reinforcing the steps taken in the scratch work helps to ensure clarity and make the proof understandable to the reader, especially when the scratch work is not shown.

Summary & Key Takeaways

• The video introduces the mathematical concept of limit and its definition.

• The presenter demonstrates the step-by-step process of proving the limit of the given function as x approaches infinity using scratch work.

• The Archimedean principle and the boundedness of the sine function are used to simplify the expression.

• The proof concludes that for any epsilon greater than zero, there exists a positive number m such that the distance between the function and zero is less than epsilon for all x greater than m.