- #1
felipe oteiza
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l'hopital must be apply, i'll be very grateful
How Does L'Hopital's Rule Apply to Lim (1/x)^tan(x) as x Approaches 0?
- Thread starter felipe oteiza
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In summary, the conversation is about the limit of x^tanx as x approaches 0. The conversation includes a clarification on the notation, a question about the limit of x^tanx, and a response with the final answer being 1. The conversation also includes a correction to the answer, clarifying that the function is actually (1/x)^tanx, which makes the limit indeterminate.
- #2
BvU
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Hello felipe 
Do you know the limit for ##\sin x \over x ## ?
Do you know the limit for ##\sin x \over x ## ?
- #3
BvU
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oops, sorry, you mean $$x^{-\tan x}\ \ ?$$
- #4
felipe oteiza
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yes! the lastBvU said:
- #5
BvU
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Where does ##\ x^{\tan x}\ ## go for ## \ x\downarrow 0 ## ?
- #6
felipe oteiza
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lim ( 1/x )^tan x as x->0BvU said:
- #7
BvU
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Yes, that was my question 
- #8
felipe oteiza
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I don't understand your questionBvU said:Yes, that was my question
(my english is not very good)- #9
BvU
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What is the limit ##\ \ \displaystyle \lim_{x\downarrow 0}
\ x^{\tan x}\ ## ?
\ x^{\tan x}\ ## ?
- #10
mathman
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tanx ~ x as x ->0, so problem can be looked at as [itex]\lim_{x-->0} x^x[/itex] However [itex]x^x=e^{xlnx}[/itex].
Since [itex]\lim_{x->0}xlnx=0[/itex], the final answer = 1.
Since [itex]\lim_{x->0}xlnx=0[/itex], the final answer = 1.
- #11
felipe oteiza
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thanksmathman said:
- #12
shaztp
Tan (0)=0 there for answer will be 1
- #13
mathman
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Not by itself. The function is [itex](\frac{1}{x})^{tanx}[/itex], so as x->0, the expression becomes [itex](\frac{1}{0})^{0}[/itex] which is indeterminate.shaztp said:
FAQ: How Does L'Hopital's Rule Apply to Lim (1/x)^tan(x) as x Approaches 0?
1. What is the limit of (1/x)^tan(x) as x approaches 0?
The limit of (1/x)^tan(x) as x approaches 0 is undefined. This is because as x approaches 0, the term (1/x) becomes very large, and the power of tan(x) becomes very small, resulting in an indeterminate form.
2. How can I determine the limit of (1/x)^tan(x) as x approaches 0?
In order to determine the limit of (1/x)^tan(x) as x approaches 0, you can use L'Hopital's rule. This rule allows you to take the derivative of the numerator and denominator separately, and then evaluate the limit again. By repeating this process, you can eventually find the limit.
3. Is there a way to simplify the expression (1/x)^tan(x) as x approaches 0?
Unfortunately, there is no way to simplify the expression (1/x)^tan(x) as x approaches 0. This is because the expression is already in its simplest form.
4. Can the limit of (1/x)^tan(x) as x approaches 0 be evaluated using a graph?
No, the limit of (1/x)^tan(x) as x approaches 0 cannot be evaluated using a graph. This is because the graph of the function has a vertical asymptote at x=0, making it impossible to determine the limit at that point.
5. Why is it important to find the limit of (1/x)^tan(x) as x approaches 0?
Finding the limit of (1/x)^tan(x) as x approaches 0 is important because it helps to understand the behavior of the function near the point x=0. It also allows us to determine the continuity of the function at that point, which is crucial in many mathematical and scientific applications.