Find the limit of sin(2x)/tan(3x) as x approaches 0

TL;DR

This video demonstrates how to use L'Hopital's Rule to solve a limit involving trigonometric functions.

Transcript

hi in this video we're going to find a limit we have the limit as X approaches 0 of the sine of 2x all divided by the tangent of 3x let's go ahead and work through this solution first thing you should do whenever you have a limit is at the very least mentally take the number and plug it in let's actually go ahead and do it if we put a zero where th... Read More

Key Insights

• ☺️ Plugging in the value of x initially to find the limit may result in an indeterminate form.
• 😑 L'Hopital's Rule can be used to simplify the expression by taking the derivatives of both numerator and denominator.
• 📏 The chain rule is applied when differentiating composite functions.
• ♾️ L'Hopital's Rule is particularly useful for handling indeterminate forms like 0/0 and infinity/infinity.
• ☺️ The limit in this video is solved by evaluating the derivatives and plugging in the value of x afterwards.
• 👨‍💼 The derivative of sine 2x is cosine 2x, and the derivative of tangent 3x is secant^2 3x.
• 🔌 After differentiation and plugging in x=0, the answer to the limit is found to be 2/3.

Questions & Answers

Q: What does plugging in zero for x initially yield?

Plugging in zero gives an indeterminate form of 0/0, which means the limit cannot be determined directly.

Q: How does L'Hopital's Rule help solve the limit?

L'Hopital's Rule states that when you have an indeterminate form like 0/0, you can take the derivative of the numerator and denominator to simplify the expression and evaluate the limit.

Q: What is the derivative of sine 2x?

The derivative of sine 2x is cosine 2x. For the chain rule, we multiply by the derivative of the inside function, giving us cosine 2x multiplied by 2.

Q: How do you compute the derivative of tangent 3x?

The derivative of tangent 3x is secant^2 3x. Using the chain rule, the derivative of the outside function is secant^2 3x, and we multiply by the derivative of the inside, which is 3.

Summary & Key Takeaways

• The video explains how to find the limit as X approaches 0 of (sin 2x) / (tan 3x) using L'Hopital's Rule.

• Initially, plugging in zero for x gives an indeterminate form of 0/0.

• L'Hopital's Rule is applied by taking the derivative of the numerator and denominator separately and evaluating the limit again.