Finding the limit of x tending to 0 of ln(x)sin(x)

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In summary, The problem asks for the limit of a function as x approaches 0. The solution involves rearranging the expression and using L'Hopital's Rule to find the limit of the product of two functions, ln(x) and sin(x)/x. The limit of sin(x)/x is known to be 1, and the limit of ln(x) as x approaches 0 from the right is negative infinity. Therefore, the limit of the original function is 0.
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knowlewj01
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Homework Statement



find the limit of:

[itex]\frac{lim}{x\rightarrow0} ln(x)sin(x)[/itex]

tip:
you know that
[itex] \frac{sin(x)}{x} \rightarrow 1[/itex]
so rearrange to use


Homework Equations



L'hopitals rule?

The Attempt at a Solution



i rearranged to get this

[itex]\frac{lim}{x\rightarrow0} \frac{\frac{ln(x)sin(x)}{x}}{\frac{1}{x}}[/itex]

i know that [itex]\frac{sin(x)}{x} \rightarrow 1[/itex]

and [itex]\frac{1}{x} \rightarrow \infty[/itex]

does [itex]ln(x) \rightarrow -\infty[/itex] ?

I might need to use L'hopitals rule after this but i don't think it will make anything any easier.
Am i missing something major here, does the fact that sin(0)=0 mean that the limit is 0? because if it is, the question is fairly misleading.
 
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  • #2
[itex]ln(x) \rightarrow -\infty[/itex]
as x --> 0+

You'll probably need to look at this as a right-side limit, since the ln function is not defined for x <=0.

Using the hint, write the limit expression as
[tex]x~ln(x) \frac{sin(x)}{x}[/tex]

The limit of a product is the product of the limits, provided that both limits in the product exist. If you rewrite x*ln(x) appropriately, you can use L'Hopital's Rule on it.
 

1. What is the limit of x tending to 0 of ln(x)sin(x)?

The limit of x tending to 0 of ln(x)sin(x) is 0.

2. How do you find the limit of x tending to 0 of ln(x)sin(x)?

To find the limit, you can use L'Hopital's rule or the squeeze theorem.

3. What is L'Hopital's rule?

L'Hopital's rule is a mathematical rule that allows you to find the limit of a function by taking the derivative of the numerator and denominator separately and then evaluating the limit again.

4. What is the squeeze theorem?

The squeeze theorem is a mathematical theorem that states if a function is always between two other functions and the two functions have the same limit at a point, then the middle function also has the same limit at that point.

5. Is it possible for the limit of x tending to 0 of ln(x)sin(x) to not exist?

Yes, it is possible for the limit to not exist if the function oscillates or approaches different values from the left and right sides of the limit point.

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