- #1
zorro
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What is wrong?
In summary, the conversation discusses the concept of rewriting a function to evaluate its limit, and how it may not always lead to a valid answer. The use of L'Hopital's rule is also mentioned as a possible solution, along with the idea of re-engineering to better understand the limit. The conversation also addresses a possible application of the quotient rule, but clarifies that it only applies if certain conditions are met.
What is wrong?
- #2
CompuChip
Science Advisor
Homework Helper
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Nothing's wrong, the fact that you can rewrite the function such that the limit becomes of the form 'infinity/infinity' when evaluated 'naively' does not mean that the limit suddenly no longer exists. In fact, this is one of the cases where we are allowed to apply L'Hopitals rule, and we then often find a finite answer.
What you could also do, of course, is rewrite it as
[tex]\lim_{n \to \infty} \frac{1 + x^n}{1 + x^n} - \frac{2 x^n}{1 + x^n}[/tex]
and then divide numerator and denominator of the second term by xn (or you can still do that first and then do the same rewriting trick, it just looks a little different).
What you could also do, of course, is rewrite it as
[tex]\lim_{n \to \infty} \frac{1 + x^n}{1 + x^n} - \frac{2 x^n}{1 + x^n}[/tex]
and then divide numerator and denominator of the second term by xn (or you can still do that first and then do the same rewriting trick, it just looks a little different).
- #3
- 13,368
- 3,518
I hope the OP sees that nothing in his limit diverges so would it need to be <reingeneered> ??
- #4
zorro
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dextercioby said:
Re-engineering helps to understand things better.
CompuChip said:
Thanks!
- #5
Landau
Science Advisor
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[tex]1=\lim_{n\to\infty}(1)=\lim_{n\to\infty}\frac{n}{n}\text{'}=\text{'}\frac{\infty}{\infty}[/tex]
You are applying the quotient rule where it is invalid to do so. The quotient rule says that IF [tex]\lim_{n\to\infty}a_n=a[/tex] and [tex]\lim_{n\to\infty}b_n=b[/tex], where a and b are real (FINITE!) numbers with b non-zero, THEN [tex]\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{a}{b}[/tex].
In this case a and b are not (finite) real numbers.
You are applying the quotient rule where it is invalid to do so. The quotient rule says that IF [tex]\lim_{n\to\infty}a_n=a[/tex] and [tex]\lim_{n\to\infty}b_n=b[/tex], where a and b are real (FINITE!) numbers with b non-zero, THEN [tex]\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{a}{b}[/tex].
In this case a and b are not (finite) real numbers.
- #6
zorro
- 1,384
- 0
Is CompuChip wrong?
FAQ: Limits Confusion: What Is Wrong?
What are limits in mathematics?
Limits in mathematics refer to the value that a function approaches as the input approaches a certain value. It can also be thought of as the value that a sequence or series approaches as the number of terms increases.
What causes confusion about limits?
The concept of limits can be confusing because it involves understanding how a function behaves near a certain point, rather than just evaluating the function at that point. This can be counterintuitive for some people and may require a deeper understanding of calculus and mathematical notation.
What are some common misconceptions about limits?
One common misconception is that the limit of a function must always be equal to the function's value at that point. However, this is not always the case as the limit may not exist or may be different from the function's value at that point.
Another misconception is that a function must approach a certain value from both sides in order to have a limit. In reality, the limit can exist even if the function only approaches the value from one side.
How can I improve my understanding of limits?
To improve your understanding of limits, it is important to practice solving limit problems and familiarize yourself with the notation and concepts. You can also seek help from a tutor or teacher if you are having trouble understanding a specific concept.
Additionally, working through real-world applications of limits, such as related rates or optimization problems, can help to solidify your understanding of the concept.
What are some resources for learning more about limits?
There are many online resources available for learning more about limits, such as Khan Academy, MathIsFun, and Paul's Online Math Notes. Your school or local library may also have textbooks or other resources on calculus and limits.
In addition, attending lectures or seeking help from a math tutor or teacher can also be beneficial in understanding limits and other calculus concepts.
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