Limit of cos(13x)/13x as x approaches infinity

TL;DR

Limit as X approaches infinity of cosine of 13X over 13X is zero, explained using intuition and the squeeze theorem.

Transcript

okay so you have to find the limit as X approaches infinity of the cosine of 13 X over 13 X so there's really two ways to do this problem so solution one is to basically just look at it and use intuition to come up with the answer so notice that when X gets really really really really big the bottom gets really really really big and the top piece c... Read More

Key Insights

• 🤩 Intuition plays a key role in understanding and solving mathematical problems.
• 👔 The squeeze theorem offers a formal method to solve limits by bounding the function.
• ⛔ Understanding the behavior of trigonometric functions is crucial in limit problems.
• ❓ Using multiple approaches can enhance problem-solving skills.
• ⛔ Squeeze theorem ensures rigor in limit evaluations.
• ⛔ Math concepts such as oscillation and bound properties are essential in limit calculations.
• 🈸 Knowing mathematical theorems and their applications can simplify complex problems.

Questions & Answers

Q: What is the intuition behind solving the limit as X approaches infinity of cosine 13X over 13X?

When X gets large, the function oscillates between -1 and 1. As the denominator (13X) grows, the function gets squeezed towards zero.

Q: How does the squeeze theorem help solve the limit in this scenario?

The squeeze theorem bounds the function between -1 and 1 and takes the limits of the bounds, showing that the function squeezed between them also approaches zero.

Q: Why is the squeeze theorem also known as the sandwich or pinching theorem?

The squeeze theorem is named so because it "squeezes" or "sandwiches" the function between two other functions whose limits are known, determining the limit of the function in question.

Q: What are the benefits of using intuition versus the squeeze theorem in solving limits?

Intuition provides a quick understanding and result, while the squeeze theorem offers a more rigorous and systematic approach, particularly useful when showing work.

Summary & Key Takeaways

• The limit as X approaches infinity of cosine 13X over 13X can be solved using intuition, where as X gets large, the function oscillates between -1 and 1, making the limit zero.

• Alternatively, the squeeze theorem can be used by sandwiching cosine 13X between -1 and 1, and taking the limits of the bounds which also approach zero, leading to the same result.

• The squeeze theorem states that if two functions approach zero and a third function is trapped between them, then the middle function also approaches zero.