Another limit using l'hopitals

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In summary: STRACT: The question asks for the limit as x approaches infinity of the expression (1/x^2) - (cscx)^2. The individual terms do not approach a finite limit as x approaches infinity, making the overall limit non-existent. Proof can be provided if necessary. In summary, the limit does not exist.
  • #1
magnifik
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Homework Statement


limit as x goes to infinity of (1/x^2) - (cscx)^2


Homework Equations





The Attempt at a Solution


I made it so the denominator is x^2, but then it would 1-inf/inf which isn't indeterminate. i need help setting it up so it would be in indeterminate form. thanks.
 
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  • #2
You probably mean lim x->0, right? Just make a common denominator and combine those two terms into a single fraction. It's probably easier to write 1/sin(x)^2 instead of csc(x)^2, but it will still take several derivatives before you get a nonindeterminant answer from l'Hopital.
 
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  • #3
nope, the question states lim x-> inf
 
  • #4
magnifik said:
nope, the question states lim x-> inf

Then tell me about the limiting behavior of 1/x^2 and csc(x)^2 as x->inf. Is that expression really indeterminant?
 
  • #5
that's what my original problem was
 
  • #6
magnifik said:
that's what my original problem was

Sketch a graph of each one. The limiting behavior should be visually obvious.
 
  • #7
magnifik said:

Homework Statement


limit as x goes to infinity of (1/x^2) - (cscx)^2


Homework Equations





The Attempt at a Solution


I made it so the denominator is x^2, but then it would 1-inf/inf which isn't indeterminate. i need help setting it up so it would be in indeterminate form. thanks.

The limit does not exist by any means. If needed, a proof can be given.

AB
 

1. What is the l'Hopital's rule?

L'Hopital's rule is a mathematical principle that allows you to evaluate limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of a quotient of two functions is indeterminate, then the limit of the quotient of their derivatives will be the same.

2. When should I use l'Hopital's rule?

You should use l'Hopital's rule when you encounter a limit that is in an indeterminate form, such as 0/0 or ∞/∞. This rule can help you solve these types of limits that may be difficult or impossible to solve using other methods.

3. What are the conditions for applying l'Hopital's rule?

The conditions for applying l'Hopital's rule are that the limit must be in an indeterminate form, and the functions in the limit must be differentiable in a neighborhood around the point where the limit is being evaluated.

4. Can l'Hopital's rule be applied multiple times?

Yes, l'Hopital's rule can be applied multiple times if the resulting limit is still in an indeterminate form. However, it is important to note that the limit may not always exist even after applying the rule multiple times.

5. Are there any limitations to using l'Hopital's rule?

Yes, there are some limitations to using l'Hopital's rule. It cannot be used to evaluate limits involving products, powers, or roots of functions. It also cannot be used for limits at infinity or for limits involving discontinuities.

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