Finding the limit without L'Hôpital's rule

In summary, the task is to prove that the limit of (1 - 1/n^2)^n as n approaches infinity is equal to 1. The first part of the question has a similar form, but uses a positive exponent instead of a negative one. The attempt at a solution involves using the generalized binomial theorem and the squeeze theorem. The relevance of the limit (1 + 1/n)^n being bounded above by e is also mentioned.
  • #1
Unredeemed
120
0

Homework Statement



Required to prove that
[itex]
\displaystyle\lim_{n\rightarrow \infty} ((1 - \frac{1}{n^2})^{n}) = 1
[/itex]

Homework Equations



[itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n})[/itex] is bounded above by e. I'm not sure if this is relevant, but it was the first part of the question, so I'd assume so?

Also, we haven't proved L'Hopital's rule yet, so I can't use that.

The Attempt at a Solution



I was thinking to maybe try and write it in a similar way to the first part.

So: [itex]
\displaystyle\lim_{n\rightarrow \infty} (((1 + \frac{1}{-(n^2)})^{-(n^2)})^{\frac{-1}{n}})
[/itex]

But, as n tends to infinity [itex]-n^2[/itex] tends to negative infinity?
 
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  • #2
Unredeemed said:

Homework Statement



Required to prove that
[itex]
\displaystyle\lim_{n\rightarrow \infty} ((1 - \frac{1}{n^2})^{n}) = 1
[/itex]

Homework Equations



[itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n})[/itex] is bounded above by e. I'm not sure if this is relevant, but it was the first part of the question, so I'd assume so?

Also, we haven't proved L'Hopital's rule yet, so I can't use that.

The Attempt at a Solution



I was thinking to maybe try and write it in a similar way to the first part.

So: [itex]
\displaystyle\lim_{n\rightarrow \infty} (((1 + \frac{1}{-(n^2)})^{-(n^2)})^{\frac{-1}{n}})
[/itex]

But, as n tends to infinity [itex]-n^2[/itex] tends to negative infinity?

Try the generalised binomial theorem. If you want to prove rigorously that all the terms except the first go to zero at the limit, use the squeeze theorem.
 
  • #3
Unredeemed said:
[itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n})[/itex] is bounded above by e. I'm not sure if this is relevant, but it was the first part of the question, so I'd assume so?

yes, it is relevant...firstly, u should understand why [itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n}) = e [/itex] then u can solve this question easily..
 

Related to Finding the limit without L'Hôpital's rule

1. What is L'Hôpital's rule and why is it used to find limits?

L'Hôpital's rule is a mathematical tool used to find limits of functions that are in indeterminate form, such as 0/0 or ∞/∞. It allows us to take the derivative of the numerator and denominator separately, making it easier to evaluate the limit.

2. Can limits be found without using L'Hôpital's rule?

Yes, limits can be found using other methods such as algebraic manipulation, factoring, or using limit laws. L'Hôpital's rule is just one method that can be used to find limits, but it is not always necessary.

3. What are some common approaches to finding limits without L'Hôpital's rule?

Some common approaches include simplifying the expression, using substitution, factoring, and using trigonometric identities. It is important to have a good understanding of algebraic concepts and limit laws in order to effectively find limits without using L'Hôpital's rule.

4. Are there any restrictions when using L'Hôpital's rule to find limits?

Yes, L'Hôpital's rule can only be used when the limit is in an indeterminate form. It also only applies to limits involving variables that approach a specific value, not to limits at infinity. Additionally, the function must be differentiable in the given interval.

5. Why is it important to know how to find limits without L'Hôpital's rule?

Understanding how to find limits without relying on L'Hôpital's rule can help build a strong foundation in calculus and strengthen problem-solving skills. It also allows for a better understanding of the behavior of functions and their limits, which is important in many areas of mathematics and science.

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