The Limit (do not use L'Hospital rule)
TLDRThe video script discusses the concept of limits in calculus, specifically focusing on the limit of sine theta over theta as theta approaches zero. The presenter clarifies the common mistake of using L'Hôpital's rule without first establishing the limit, which could lead to circular reasoning. Instead, the correct approach is demonstrated using geometrical methods, starting with the unit circle and forming a right triangle to derive the relationship between the arc length, the tangent line, and the sine of theta. The presenter then uses inequalities and the squeeze theorem to show that the limit equals one for both approaches as theta goes to zero from the positive and negative directions. The video serves as a caution against mathematical shortcuts and emphasizes the importance of understanding the underlying principles.
Takeaways
- 🔢 The limit of sine theta over theta as theta approaches 0 is a common mathematical problem that can be solved using L'Hôpital's rule, but it's important to first establish this limit through geometric reasoning to avoid circular logic.
- 📐 Geometry is used to solve the limit by considering the unit circle and the behavior of the sine function as the angle approaches 0.
- 📈 As theta approaches 0, the arc length on the unit circle approaches the angle in radians, which is a key step in establishing the limit.
- 🧍♂️ A right triangle is formed on the unit circle, with the sine of theta representing the opposite side, the radius (which is 1) as the hypotenuse, and the tangent line representing the adjacent side.
- 📉 The inequality sine theta < theta < tangent theta is established, which is crucial for applying the squeeze theorem.
- 🔄 By taking the reciprocal of the inequality and multiplying through by sine theta, the inequality is transformed to show that 1 < sine theta over theta < cotangent theta.
- 📌 The squeeze theorem is applied to conclude that the limit of sine theta over theta as theta approaches 0 is 1.
- ➡️ The process is repeated for the left-handed limit (theta approaching 0 from the negative side) to ensure the limit holds true for both directions.
- 🤔 The script highlights the importance of showing work and understanding the underlying principles, rather than just applying formulas without proof.
- ❌ The video warns against the pitfalls of circular reasoning, emphasizing the need to establish fundamental limits before using them in proofs.
- 📚 The script provides a detailed walkthrough of a mathematical proof, which is valuable for educational purposes and understanding the depth of mathematical concepts.
- 💯 The video concludes by emphasizing the importance of understanding the process and not just the result, which is a valuable lesson for students and mathematicians alike.
Q & A
What is the limit as Delta (theta) approaches 0 of sine theta over theta?
-The limit as theta approaches 0 of sine theta over theta is 1. This is derived using geometrical arguments and the squeeze theorem, which shows that the limit is bounded between 1 and 1 as theta approaches 0 from both the positive and negative directions.
Why does the script mention L'Hôpital's rule but then decide to use geometry instead?
-L'Hôpital's rule is initially considered because the expression results in an indeterminate form of 0/0 as theta approaches 0. However, the script chooses geometry to avoid circular reasoning, since knowing the limit is a prerequisite for differentiating sine theta and obtaining cosine theta.
How does the unit circle help in finding the limit of sine theta over theta as theta approaches 0?
-The unit circle is used to create a right triangle where the angle theta is small. The arc length on the unit circle corresponding to angle theta represents the angle's measure, and the sides of the right triangle formed can be related to the sine and tangent of theta, which helps in establishing inequalities that lead to the limit.
What is the significance of the squeeze theorem in this context?
-The squeeze theorem is used to conclude that the limit of sine theta over theta as theta approaches 0 must be 1. It is applied after establishing that the expression is bounded between 1 and 1 for theta approaching 0 from both directions.
Why is it important to consider both the positive and negative approaches to 0 for theta?
-Considering both the positive and negative approaches to 0 for theta is important to ensure the limit exists and is the same from both sides, which is a requirement for the limit to be defined at theta equals 0.
How does the script handle the case when theta approaches 0 from the negative side?
-The script uses the property of sine being an odd function, which means sine(-theta) = -sine(theta). By considering the negative of sine theta over negative theta, the expression simplifies to sine theta over theta, which is the same as the positive approach case, thus showing the limit is the same from both sides.
What is the role of the tangent line in the geometric argument?
-The tangent line is used to form a right triangle where the tangent of theta is the ratio of the opposite side to the adjacent side. This helps in establishing the inequality between the tangent theta and sine theta, which is crucial for applying the squeeze theorem.
Why is it considered bad practice to use the derivative to prove the limit without first establishing the limit itself?
-Using the derivative to prove the limit without first establishing the limit is an example of circular reasoning. It is logically flawed because the derivative is calculated based on the assumption that the limit exists, which is what is being attempted to prove.
What does the script mean by 'circular reasoning'?
-Circular reasoning is a logical fallacy where the conclusion of an argument is derived using the conclusion itself as a premise or where the argument's premises are assumed to be true because the conclusion is true. In the context of the script, it refers to using the derivative of sine theta to prove the limit of sine theta over theta without first proving the limit independently.
How does the inequality involving sine theta, the arc length, and tangent theta help in finding the limit?
-The inequality shows that sine theta is less than the arc length (which is theta) and less than tangent theta. By taking the reciprocal of these expressions and applying the squeeze theorem, it is possible to show that the limit of sine theta over theta as theta approaches 0 is bounded between 1/1 (which is 1) and 1, hence the limit must be 1.
What is the final conclusion about the limit of sine theta over theta as theta approaches 0?
-The final conclusion is that the limit of sine theta over theta as theta approaches 0 is 1. This is derived from geometric arguments and the application of the squeeze theorem, considering both the positive and negative approaches to 0 for theta.
Outlines
📚 Introduction to Limit Calculations
The video begins by discussing the limit of sine theta over theta as theta approaches zero. The presenter uses L'Hopital's rule to differentiate sine and obtain cosine theta. They emphasize the importance of showing work and avoiding circular reasoning. The presenter also shares a personal anecdote about getting zero points on a test due to incomplete work.
📐 Geometric Approach to Limit Calculation
The presenter explains the geometric approach to finding the limit of sine theta over theta as theta approaches zero from the positive side. They start by considering the unit circle and drawing a right triangle with angle theta. The presenter uses the arc length formula and the Pythagorean theorem to derive inequalities involving sine theta, arc length theta, and tangent theta. They then apply the squeeze theorem to conclude that the limit is equal to one.
🔄 Left-Hand Limit and Final Conclusion
The presenter briefly discusses the left-hand limit as theta approaches zero from the negative side. They use the property of sine being an odd function to simplify the expression and show that the left-hand limit is also equal to one. With both the left and right limits being equal, the presenter concludes that the overall limit as theta approaches zero of sine theta over theta is one. They reiterate the importance of understanding this concept for calculus students and invite viewers to share their experiences and ask questions.
Mindmap
Keywords
💡Limit
💡Delta
💡L'Hôpital's Rule
💡Derivative
💡Unit Circle
💡Arc Length
💡Right Triangle
💡Hypotenuse
💡Sine Theta
💡Cosine Theta
💡Squeeze Theorem
Highlights
The limit as Delta (theta) goes to 0 of sine theta over theta results in an indeterminate form 0/0.
L'Hôpital's rule is applied to resolve the indeterminate form, leading to the conclusion that the limit is cosine of 0, which is 1.
The importance of showing work and avoiding circular reasoning when using derivatives is emphasized.
Geometry is used to find the limit of sine theta over theta as theta approaches 0, avoiding the need for prior knowledge of the derivative of sine theta.
The unit circle is introduced to visualize the problem, focusing on the first quadrant for small angles.
The arc length formula is used, with the radius being 1 (unit circle) and the angle being theta, resulting in an arc length of theta.
A right triangle is formed, with the hypotenuse being 1 and the opposite side being sine theta, establishing a relationship between the sides.
The tangent line to the unit circle at (1,0) is used to form another right triangle, leading to the relationship tangent theta equals 1/cosine theta.
Inequalities are established between sine theta, the arc length theta, and tangent theta, with theta approaching 0.
The reciprocal of the inequalities is taken, leading to a relationship between 1/sine theta, 1/theta, and cotangent theta.
By multiplying through by sine theta, the inequality simplifies to show that cosine theta is the limit of sine theta over theta as theta approaches 0.
The squeeze theorem is applied to conclude that the limit of sine theta over theta as theta approaches 0 is indeed 1.
The process is repeated for the left-hand limit as theta approaches 0, showing that sine theta is an odd function and the limit remains the same.
The final conclusion is that the limit of sine theta over theta as theta approaches 0 from both the right and left is 1, avoiding the issue of 0/0.
The video addresses common mistakes and misunderstandings regarding limits and derivatives, providing a clear and rigorous approach to the problem.
The presenter encourages viewers to share their experiences with similar problems and to engage with the content through comments.
The video concludes with an invitation to subscribe for more math content, emphasizing the educational value of the channel.
Transcripts
Browse More Related Video
Proof: Limit of sinx/x as x approaches 0 with Squeeze Theorem | Calculus 1
Limit of sin(x)/x as x approaches 0 | Derivative rules | AP Calculus AB | Khan Academy
Calculus 1: Finding Limits of Trigonometric Functions | Math with Professor V
Squeeze Theorem for Limits (1)
Trig limit using pythagorean identity | Limits and continuity | AP Calculus AB | Khan Academy
Proof: lim (sin x)/x | Limits | Differential Calculus | Khan Academy
5.0 / 5 (0 votes)
Thanks for rating: