Limits at infinity of quotients with trig | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

The limit as X approaches infinity of cosine X over X squared minus one is zero.

Transcript

• [Voiceover] So, let's see if we can figure out what the limit as X approaches infinity of cosine of X over X squared minus one is. And like always, pause this video and see if you can work it out on your own. Well, there's a couple of ways to tackle this. You could just reason through this and say, "Well, look this numerator, right over here, cos... Read More

Key Insights

• ☺️ The numerator, cosine X, oscillates between negative one and one as X increases.
• ⬛ The denominator, X squared minus one, becomes infinitely large as X increases.
• 🌥️ Dividing a bounded numerator by an infinitely large denominator results in the limit approaching zero.
• 🥹 This limit holds true for all X approaching infinity.
• ☺️ The limit of cosine X over X squared minus one as X approaches infinity is between zero.
• ⛔ The limit can be expressed as zero is less than or equal to the limit is less than or equal to zero.
• ⛔ Mathematically bounding the numerator confirms the limit as zero.

Questions & Answers

Q: How does the numerator, cosine X, behave as X increases towards infinity?

The numerator, cosine X, oscillates between negative one and one as X increases towards infinity. Its value alternates between these two limits.

Q: What happens to the denominator, X squared minus one, as X approaches infinity?

The denominator, X squared minus one, grows infinitely large as X approaches infinity. This is because the X squared term dominates the subtraction of one.

Q: How can we determine the limit of cosine X over X squared minus one as X approaches infinity?

We can reason that the numerator stays bounded between negative one and one, while the denominator becomes infinitely large. Dividing a bounded numerator by an infinitely large denominator yields a limit of zero.

Q: Is there a more mathematical way to approach this problem?

Yes, we can mathematically bound the numerator. By stating that cosine X ranges from negative one to one, we can say that cosine X over X squared minus one is less than or equal to one over X squared minus one and greater than or equal to negative one over X squared minus one.

Summary & Key Takeaways

• The numerator, cosine X, oscillates between negative one and one as X increases.

• The denominator, X squared minus one, becomes infinitely large as X increases.

• As a result, the limit of cosine X over X squared minus one as X approaches infinity is zero.