Limits: Solving x^3e^(-x^2) with L'Hospital's Rule

  • Thread starter grothem
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In summary, the conversation is discussing the limit of x^3 * e^(-x^2) as x approaches infinity. The speaker mentions using L'Hospital's rule three times, but is having trouble getting the correct answer. They suggest rewriting e^(-x^2) using negative exponents and using L'Hospital's rule again. The other speaker suggests using the product rule and taking the derivative of e^(-x^2). Ultimately, the limit is determined to be 0 due to the growth rate of e^(-x^2) being much faster than x^3.
  • #1
grothem
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Lim x[tex]\rightarrow[/tex][tex]\infty[/tex] x^3 [tex]\times[/tex] e^(-x^2)

I moved x^3 down to make it a fraction and then I used L'Hospital's rule, but I can't come up with the right answer. Not sure what I'm doing wrong
 
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  • #2
that looks kind of confusing...

its the limit of x^3 * e^(-x^2) as x approaches infinity
 
  • #3
well, i guess this is going to be zero, since [tex] e^{x^2}[/tex] grows much faster than x^3
 
  • #4
What do you mean you "moved it down"?

Regardless, I suggest rewriting [itex]e^{-x^{2}}[/itex] using the facts you know about negative exponents. Probably easier that way.
 
  • #5
and if you want to use l'hopitals rule, you need to do it three times from what i can see!
 
  • #6
ah ok...I used l'hospital's rule 3 times and ended up with

lim (3/(4x^3*e^(x^2)) = 0
 
  • #7
grothem said:
ah ok...I used l'hospital's rule 3 times and ended up with

lim (3/(4x^3*e^(x^2)) = 0

Well the bottom will defenitely not be what u got!
[e^(x^2)]'=(x^2)'e^(x^2)]=[2xe^(x^2)] not take again the derivative and apply the product rule

(uv)'=u'v+v'u
 

Related to Limits: Solving x^3e^(-x^2) with L'Hospital's Rule

1. What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical theorem that can be used to evaluate the limit of a function that has an indeterminate form, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions is indeterminate, then the limit of the ratio of their derivatives will be the same.

2. How do I know when to use L'Hospital's Rule?

L'Hospital's Rule should only be used when the limit of a function has an indeterminate form, such as 0/0 or ∞/∞. It is important to note that this rule should only be used as a last resort after other methods, such as direct substitution, have been attempted.

3. What is the process for using L'Hospital's Rule?

The process for using L'Hospital's Rule involves taking the derivative of both the numerator and denominator of the function and then evaluating the limit again. This process can be repeated until the limit is no longer indeterminate.

4. Can L'Hospital's Rule be used for all types of limits?

No, L'Hospital's Rule can only be used for limits that have an indeterminate form, such as 0/0 or ∞/∞. It cannot be used for limits that have a different form, such as ∞-∞ or 1^∞.

5. Are there any limitations to using L'Hospital's Rule?

Yes, there are limitations to using L'Hospital's Rule. It should only be used for limits that have an indeterminate form and the functions involved should be differentiable. Additionally, it should only be used as a last resort after other methods have been attempted.

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