Limit of (1+sin(4x))^(cot(x)) as x goes to 0+, L'Hospital's Rule

TL;DR

This content discusses finding the limit of a complex function by using the properties of cotangent and Ln.

Transcript

I would like to work out example number eight in section 4.4 with you guys the limit when X is appro to 0 + 1 + S of 4X raised to the cotangent X power all right we are plugging zero into all the X here we'll get 1+ S of 0 which is 0 1 + 0 is 1 raise to the pluging Z into Cent X cotangent 0 plus it's going to be Infinity all right so let me make a ... Read More

Key Insights

• 🔌 Plugging zero into the cotangent function results in positive infinity.
• 👻 The introduction of a new variable allows for the simplification of the equation.
• ❓ The Ln property is applied to the equation to further simplify the problem.
• 📏 L'Hôpital's rule can be used when both the numerator and denominator approach zero.
• ☺️ By rewriting cotangent X as 1 over cotangent X, the equation can be further manipulated.
• 🥡 Taking the derivative of the Ln function helps in finding the limit of the original function.
• ❓ The final answer is obtained by using the properties of Ln and exponential function.

Questions & Answers

Q: What is the limit of the function as X approaches 0+?

The limit is positive infinity when X approaches 0+ and cotangent is involved.

Q: How does introducing a new variable help in solving the problem?

Introducing a new variable allows the speaker to manipulate the equation and apply the Ln property, simplifying the problem.

Q: What happens when the numerator and denominator both approach zero?

When both the numerator and denominator approach zero, the limit can be determined using l'Hôpital's rule.

Q: How does the speaker simplify the cotangent function?

The speaker rewrites cotangent X as 1 over cotangent X, which allows for easier manipulation of the equation.

Summary & Key Takeaways

• The speaker works through example number eight in section 4.4, finding the limit of a function as X approaches 0+.

• When X is approaching 0+ and cotangent is involved, the limit is positive infinity.

• To simplify the problem, the speaker introduces a new variable and uses the Ln property.

• By manipulating the equation and applying the Ln property, the speaker is able to simplify the problem further.