Proof: lim (sin x)/x | Limits | Differential Calculus | Khan Academy

TL;DR

Using the squeeze theorem and visual trigonometry, the limit of sine of x over x as x approaches 0 is equal to 1.

Transcript

Now that we have hopefully a decent understanding of the squeeze theorem, we'll use that to prove that the limit-- I'll do it in yellow-- the limit as x approaches 0 of sine of x over x is equal to 1. And you must be bubbling over with anticipation now, because I've said this so many times. So let's do it, and actually, we have to go with-- obvious... Read More

Key Insights

• ☺️ The limit of sine of x over x as x approaches 0 is proven to be equal to 1 using the squeeze theorem.
• 😒 Visual trigonometry helps establish an inequality that allows for the use of the squeeze theorem.
• 👨‍💼 The concept of the unit circle and the definitions of sine and tangent are important in understanding the proof.
• 🧀 The proof involves calculating the areas of triangles and a wedge on the unit circle.
• ☺️ The squeeze theorem guarantees that as x approaches 0, sine of x over x approaches 1.
• 😑 The proof accounts for all quadrants by taking the absolute value of the trigonometric expressions.
• ☺️ The concept of absolute value ensures that the inequality holds for both positive and negative x values.

Questions & Answers

Q: How does visual trigonometry help prove the limit of sine of x over x?

Visual trigonometry helps in understanding the relationships between angles, sides of triangles, and the unit circle. By visualizing the unit circle and the properties of trigonometric functions, we can derive an inequality and prove the limit.

Q: Why is the absolute value of sine of x over x always positive?

In the first and fourth quadrants, where we are taking the limit as x approaches 0, sine of x is positive and x is positive or negative. The division of two positive numbers or two negative numbers always results in a positive value.

Q: What is the role of the squeeze theorem in this proof?

The squeeze theorem is used to show that as x approaches 0, the function sine of x over x is "squeezed" between two functions: 1 and cosine of x. Since both 1 and cosine of x approach 1 as x approaches 0, the squeeze theorem guarantees that sine of x over x also approaches 1.

Q: Can you explain the concept of the squeeze theorem in general?

The squeeze theorem applies when a function is "squeezed" between two other functions whose limits are equal. If the squeezed function is always in between and approaches the same limit as the squeezing functions, then the squeezed function also approaches that limit.

Summary & Key Takeaways

• The video provides a proof of the limit as x approaches 0 of sine of x over x, using the squeeze theorem and visual trigonometry.

• It begins by explaining the concept of the unit circle and the definitions of sine and tangent.

• The proof involves calculating the areas of triangles and a wedge on the unit circle to establish an inequality that leads to the limit being equal to 1.