Zero-Bounded Limit Theorem (with example)
TLDRIn this video, Dave Johnson introduces the Zero-Bounded Limit Theorem, a powerful and often overlooked limit theorem in calculus. The theorem is particularly useful in situations where other methods like L'Hôpital's rule fail. Johnson demonstrates its application through an example involving the limit as X approaches infinity of (e^X + sin(X)) over (e^X + cos(X)). He explains that by factoring out e^X from both the numerator and denominator, the indeterminate form is resolved. Applying the Zero-Bounded Limit Theorem, Johnson shows that since e^X approaches zero and both sin(X) and cos(X) are bounded functions, their respective products with e^X approach zero. Consequently, the overall limit simplifies to 1, providing a clear and concise solution without resorting to differentiation.
Takeaways
- 📚 The Zero-Bounded Limit Theorem is a powerful theorem in calculus that can be applied in situations where other methods, like L'Hôpital's rule, fail.
- 🔍 The theorem is not always included in calculus textbooks, but it is simple to apply and can be very effective for certain limit problems.
- ⚖️ The theorem states that if a function B(x) is bounded on an interval around a point 'a' and another function G(x) has a limit of zero as x approaches 'a', then the product B(x)G(x) approaches zero as x approaches 'a'.
- 📈 The function G(x) does not need to be defined at 'a', it just needs to have a limit at 'a'. The point 'a' can even be infinity.
- 🚫 L'Hôpital's rule cannot be used on every limit problem. The example in the script demonstrates a case where L'Hôpital's rule does not help in solving the limit.
- 🌟 The concept of a function being bounded means that it does not go to positive or negative infinity within a given interval.
- ✅ An example problem is provided where the limit as x approaches infinity of (e^x + sinx) / (e^x + cosx) is solved using the Zero-Bounded Limit Theorem.
- 📉 The function e^x * sinx and e^x * cosx both approach zero as x approaches infinity because e^x approaches zero and sinx and cosx are bounded functions.
- 🔢 The range of both sinx and cosx is from -1 to 1, which is why they are considered bounded functions.
- 📌 Factoring out e^x from the numerator and denominator helps to eliminate the indeterminate form and apply the Zero-Bounded Limit Theorem.
- 🎓 The final answer to the example problem is 1, which is obtained after applying the theorem and calculating the respective limits.
Q & A
What is the Zero Bounded Limit Theorem?
-The Zero Bounded Limit Theorem is a limit theorem that allows you to find the limit of a product of two functions as x approaches a certain value. Specifically, if a function B(x) is bounded on an interval and another function G(x) has a limit of zero as x approaches a, then the product B(x) * G(x) will also have a limit of zero at that point.
What are the conditions for the Zero Bounded Limit Theorem to apply?
-The Zero Bounded Limit Theorem applies if two conditions are met: (1) a function B(x) is bounded on an interval around a certain point 'a', and (2) another function G(x) has a limit of zero as x approaches 'a'.
Why is the Zero Bounded Limit Theorem considered powerful?
-The Zero Bounded Limit Theorem is considered powerful because it can be applied in situations where other theorems might not work. It is particularly useful when dealing with limits that involve infinity or are otherwise indeterminate.
What does it mean for a function to be bounded?
-A function is said to be bounded if it does not approach positive or negative infinity on a given interval. It means that the function's values stay within a certain range and do not go to extremes.
Why can't L'Hôpital's Rule be used to solve the example limit provided in the script?
-L'Hôpital's Rule cannot be used in the example because applying it results in another indeterminate form of infinity over infinity. The derivative of the numerator and denominator does not simplify the expression to a determinate form that can be evaluated.
How does the script demonstrate the application of the Zero Bounded Limit Theorem to the example limit?
-The script demonstrates the application by factoring out e^x from both the numerator and denominator, which eliminates the indeterminate form. It then applies the Zero Bounded Limit Theorem to the remaining terms, noting that the limit of e^x * sin(x) and e^x * cos(x) as x approaches infinity is zero, since e^x approaches zero and sin(x) and cos(x) are bounded.
What is the final result of the limit example provided in the script?
-The final result of the limit as x approaches infinity of (e^x + e^x * sin(x)) / (e^x + e^x * cos(x)) is 1, after applying the Zero Bounded Limit Theorem and simplifying the expression.
Why are sin(x) and cos(x) considered bounded functions?
-Sin(x) and cos(x) are considered bounded functions because their range of values is always between -1 and 1 for all real numbers x. This means they never approach infinity or negative infinity.
What is the significance of the limit of e^x as x approaches infinity?
-The limit of e^x as x approaches infinity is significant because it approaches infinity. However, when multiplied by a function that approaches zero, such as sin(x) or cos(x), the overall limit of the product can be determined using the Zero Bounded Limit Theorem to be zero.
What does it mean for a limit to be undefined?
-A limit is said to be undefined when the function does not approach a specific value or infinity as the input (in this case x) approaches a certain point. For example, the limit of sin(x) as x approaches infinity is undefined because it does not approach a specific value; it oscillates between -1 and 1.
Why is it recommended to attempt solving limits algebraically before using L'Hôpital's Rule?
-It is recommended to attempt algebraic solutions first because it helps build a strong foundation in understanding limits and their behavior without relying on differentiation. This practice is also beneficial for those who have not yet learned differentiation or L'Hôpital's Rule.
How does the script illustrate the process of factoring out e^x from the numerator and denominator?
-The script illustrates this by showing that e^x can be factored out from both the numerator (as e^x * sin(x)) and the denominator (as e^x * cos(x)). This factorization simplifies the expression and eliminates the indeterminate form, making it easier to apply the Zero Bounded Limit Theorem.
Outlines
📘 Introduction to the Zero Bounded Limit Theorem
Dave Johnson introduces the Zero Bounded Limit Theorem, a lesser-known limit theorem in calculus. He emphasizes its power and simplicity, noting that it can be applied in situations where other theorems fail. The theorem is stated without proof, but it is mentioned that it can be proven using the squeeze theorem. The theorem is applicable to functions that are bounded on an open interval around a certain point 'a', and it is particularly useful when dealing with the product of a bounded function and a function that tends to zero as it approaches 'a'. The video aims to demonstrate the theorem's application through an example that cannot be solved using L'Hôpital's rule.
🔍 Applying the Zero Bounded Limit Theorem to an Indeterminate Form
The video script presents an example involving an indeterminate form of the type ∞/∞. Dave attempts to solve this limit using algebraic manipulation rather than differentiation. He factors out e^x from both the numerator and the denominator, transforming the indeterminate form into a more manageable expression. The resulting limit is then separated into two parts: the limit of (1 + e^x * sin(x)) as x approaches infinity and the limit of (1 + e^x * cos(x)) as x approaches infinity. It is shown that applying L'Hôpital's rule does not help in this case, and a different approach is necessary. The zero bounded limit theorem is applicable here because e^x tends to zero as x approaches negative infinity, and both sin(x) and cos(x) are bounded functions.
🎓 Conclusion and Final Answer Using the Zero Bounded Limit Theorem
Dave concludes the example by applying the zero bounded limit theorem to the separated parts of the limit. Since sin(x) and cos(x) are bounded between -1 and 1, and the limit of e^x as x approaches infinity is zero, the limits of both e^x * sin(x) and e^x * cos(x) are zero. Consequently, the overall limit of the original expression simplifies to 1/1, which equals one. The video ends with a recap of the process and an encouragement for viewers to like, subscribe, and look forward to the next video.
Mindmap
Keywords
💡Zero Bounded Limit Theorem
💡Indeterminate Form
💡Bounded Function
💡Limit
💡Squeeze Theorem
💡L'Hôpital's Rule
💡Factoring
💡E to the X
💡Sin(x) and Cos(x)
💡Infinity
💡Algebraic Approaches
Highlights
Dave Johnson introduces the Zero Bounded Limit Theorem, a powerful limit theorem not commonly found in all calculus textbooks.
The theorem is useful in situations where other theorems like L'Hôpital's rule might not apply.
The Zero Bounded Limit Theorem states that if a function B is bounded on an interval and another function G has a limit of zero as X approaches a, then the product B*G approaches zero.
G of X does not need to be defined at x equals a, only that it has a limit at x as a.
The theorem is illustrated using an example where L'Hôpital's rule cannot be applied.
The example involves the limit as X approaches infinity of (e^x + sinx) over (e^x + cosx).
Factoring out e^x from the numerator and denominator eliminates the indeterminate form.
The limit of e^x * sinx as X approaches infinity is zero due to sinx being bounded and e^x approaching zero.
Similarly, the limit of e^x * cosx as X approaches infinity is also zero for the same reasons.
Sinx and cosx are bounded functions because their range is between -1 and 1.
By applying the Zero Bounded Limit Theorem, both the limits of e^x * sinx and e^x * cosx are determined to be zero.
The final answer to the example is one, obtained by evaluating the simplified limit expressions.
The video emphasizes the importance of using algebraic approaches to solving limits before resorting to L'Hôpital's rule.
Dave Johnson encourages viewers to get into the habit of using algebraic methods for solving limits.
The video demonstrates that L'Hôpital's rule is not universally applicable to all limit problems.
The Zero Bounded Limit Theorem provides an alternative method for solving limits that can be easier to apply in certain cases.
The theorem is applicable even when the function is not continuous, as long as it is bounded.
The video concludes by reinforcing the value of the Zero Bounded Limit Theorem and its practical application in solving limits.
Transcripts
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