Limits of trigonometric functions | Limits and continuity | AP Calculus AB | Khan Academy | Summary and Q&A

TL;DR

The video explains how to find limits of trigonometric functions and when limits do not exist.

Key Insights

• 😥 Sine and cosine functions have defined limits for all real numbers and the limit at a defined point is equal to the value of the function at that point.
• ⛔ The limit of tangent and cotangent functions may not exist when the limit point is not in their respective domains.

Transcript

Read and summarize the transcript of this video on Glasp Reader (beta).

Questions & Answers

Q: What is the limit as x approaches pi of sine x?

The limit is equal to the value of sine of pi, which is 0. Sine and cosine functions are continuous over their entire domain, so taking the limit at a defined point gives the function value at that point.

Q: How do we find the limit as x approaches pi/4 of cosine x?

The limit is equal to the value of cosine of pi/4, which is √2/2. Similar to sine, cosine is defined and continuous for all real numbers, so taking the limit at a defined point gives the function value at that point.

Q: What is the limit as x approaches pi of tangent x?

The limit is equal to 0 because both sine of pi and cosine of pi are defined, and there is no zero in the denominator. Tangent is defined for all real numbers except for pi/2.

Q: Why does the limit as x approaches pi/2 of tangent x not exist?

The limit does not exist because substituting pi/2 leads to 1/0, which is undefined. Pi/2 is not in the domain of tangent, so there is a vertical asymptote at that point.

Q: What is the limit as x approaches pi of cotangent x?

The limit does not exist because substituting pi leads to -1/0, which is undefined. Pi is not in the domain of cotangent, so there is a vertical asymptote at that point.

Summary & Key Takeaways

• Sine and cosine functions are defined and continuous for all real numbers, making their limits equal to the function value at that point.

• Tangent function is defined for all real numbers except for pi/2, so the limit at pi/2 does not exist.

• Cotangent function is defined for all real numbers except for pi, so the limit at pi does not exist.