Calculating the Limit Using L'Hopital's Rule and Exponential Properties

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In summary, the conversation is about finding the limit of a tricky function using L'Hopital's rule. The speaker suggests using Taylor Polynomials approximation, but the others explain how to solve it using the properties of the derivatives of arcsin. The final result is that the limit tends to 1/6.
  • #1
gipc
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I know I should apply L'Hopital's rule and use a^b=e^(b*ln(a)) but I can't finish the calculations.

limit as x->0 ((arcsin(x))/x) ^(1/x^2)
 
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  • #2
thats a tricky one, so going with what you said
[tex] \lim_{x \to 0} (\frac{arcsin(x)}{x})^{\frac{1}{x^2}}
= \lim_{x \to 0} e^{\frac{ln(\frac{arcsin(x)}{x})}{x^2}}[/tex]

now let
[tex] b =\frac{ln(\frac{arcsin(x)}{x})}{x^2}[/tex]

if the limit exists, its equal to e^(b), so finding the limit of a is sufficient

thats 0/0 indeterminate, so we can apply L'Hops rule - though i can see it will be a bit messy
 
Last edited:
  • #4
sorry, still a no-go. can't get the algebra together. can someone please help? I've applied L'hopital's rule 3 times and it keeps getting uglier.
 
  • #5
I think it might be helpful considering Taylor Polynomials approximation.
 
  • #6
We didn't learn yet the Taylor thingie. This assignment is about L'Hopital's rule.
 
  • #7
ok, so how about starting by looking at the arcsin function and its derivatives, let's abuse the notation a bit and call it a for brevity recognising its a function of x:
[tex]a(x) = arcsin(x), \ \ \ \ \ \ \lim_{x \to 0} a(x) = 0 [/tex]
[tex]a'(x) = (1-x^2)^{1/2}, \ \ \ \ \lim_{x \to 0} a'(x) = 1 [/tex]
[tex]a''(x) = x(1-x^2)^{3/2}, \ \ \ \lim_{x \to 0} a''(x) = 0 [/tex]
[tex]a'''(x) = (1-x^2)^{3/2} -3x(1-x^2)^{5/2}, \ \lim_{x \to 0} a'''(x) = 0 [/tex]
 
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  • #8
now going back to
[tex] b = \lim_{x \to 0}\frac{ln(\frac{arcsin(x)}{x})}{x^2}
= \lim_{x \to 0} \frac{ln(a) - ln(x)}{x^2}
[/tex]

this is 0/0 so using L'Hop
[tex]
= \lim_{x \to 0} \frac{a'/a - 1/x}{x^2} = \lim_{x \to 0}\frac{1}{2} \frac{a'x - a}{ax^2}
[/tex]

once again, this is 0/0 so using L'Hop
[tex]
= \lim_{x \to 0}\frac{1}{2} \frac{a''x}{a'x^2+ 2ax}= \lim_{x \to 0}\frac{1}{2} \frac{a''}{a'x+ 2a}
[/tex]

one more time, this is 0/0 so using L'Hop
[tex]
= \lim_{x \to 0}\frac{1}{2} \frac{a'''}{a''x+a'+ 2a'}= \lim_{x \to 0}\frac{1}{2} \frac{a'''}{a''x+3a'}
[/tex]

and at this point you should be able to sub in with the properties of the derivatives
 

What is L'Hopital's Rule and how is it used to calculate limits?

L'Hopital's Rule is a mathematical principle used to evaluate limits of functions that are indeterminate forms. It states that the limit of a quotient of two functions, both approaching zero or infinity, is equal to the limit of the quotient of their derivatives. This rule is particularly useful when dealing with limits involving exponential functions.

What are the steps to applying L'Hopital's Rule to calculate a limit?

The steps to applying L'Hopital's Rule are as follows:
1. Determine if the limit is in an indeterminate form (ex. 0/0, ∞/∞).
2. Take the derivative of the numerator and denominator separately.
3. Simplify the resulting expression.
4. Evaluate the limit using the simplified expression.
5. If the limit is still in an indeterminate form, repeat the process until it can be evaluated.

Can L'Hopital's Rule be used for limits involving other types of functions besides exponential functions?

Yes, L'Hopital's Rule can be applied to limits involving any type of function as long as the limit is in an indeterminate form. However, it is most commonly used for limits involving exponential functions.

Are there any restrictions or conditions for using L'Hopital's Rule to calculate a limit?

Yes, there are some restrictions and conditions for using L'Hopital's Rule:
1. The limit must be in an indeterminate form.
2. The functions involved must be differentiable.
3. The limit must approach either 0 or infinity.
4. The limit must be a one-sided limit.

Are there any alternative methods for calculating limits besides L'Hopital's Rule?

Yes, there are other methods for calculating limits, such as using the properties of limits, algebraic manipulation, and graphing. However, L'Hopital's Rule is often the most efficient and straightforward method for evaluating limits involving exponential functions.

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