Finding an Indeterminate Limit with L'Hôpital's Rule: Help and Explanation

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In summary, the conversation discusses a limit problem involving an indeterminate form of ∞ * 0. The person tried to apply L'Hôpital's rule but ended up with an expression of the form x2 rx. They are now seeking help to solve the problem correctly.
  • #1
swampwiz
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I can't seem to figure out how to find this seemingly simple limit (that is shown numerically to go to 0)

lim x → ∞ ( x rx )

for r < 1

This is an indeterminate form of ∞ * 0, but when I try to apply L'Hôpital's rule as

lim x → ∞ ( rx / ( 1 / x ) )

I end up getting an expression of the form x2 rx, with further application of the rule generating higher and higher powers of x

I'm totally stuck!
 
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  • #2
You have the right idea. Try this instead:

lim x → ∞ (x/(1/r)x)
 
  • #3
gb7nash said:
You have the right idea. Try this instead:

lim x → ∞ (x/(1/r)x)

But wouldn't that be ∞ / ∞ ? L'Hopital's rule only applies to 0 / 0
 
  • #4
swampwiz said:
L'Hopital's rule only applies to 0 / 0

No it doesn't. It also applies to +- inf/inf
 

What is L'Hôpital's Rule?

L'Hôpital's Rule is a mathematical theorem that allows us to find the limit of a function that is in an indeterminate form (such as 0/0 or ∞/∞). It states that if the limit of the quotient of two functions approaches an indeterminate form, the limit of the original function can be found by taking the quotient of the derivatives of the two functions.

When do we use L'Hôpital's Rule?

L'Hôpital's Rule is used when we encounter an indeterminate form in a limit problem. This can happen when we have a function with a variable in the denominator, or when we have a limit of a function raised to a power.

What are the steps for using L'Hôpital's Rule?

The steps for using L'Hôpital's Rule are as follows:

  1. Determine if the limit is in an indeterminate form (0/0 or ∞/∞).
  2. If it is, take the derivative of the numerator and denominator of the original function.
  3. Plug these derivatives into the limit and simplify.
  4. If the new limit is still in an indeterminate form, repeat the process until the limit is no longer indeterminate.

Can L'Hôpital's Rule be used for all limits?

No, L'Hôpital's Rule can only be used for limits that approach an indeterminate form. If the limit approaches a finite number, the rule cannot be applied.

Are there any limitations or exceptions to L'Hôpital's Rule?

Yes, there are some limitations and exceptions to L'Hôpital's Rule. It cannot be used for limits that approach zero or infinity at the same time, or for limits involving trigonometric functions. Additionally, the rule may not work if the function is not differentiable or if the derivatives of the numerator and denominator do not exist.

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