I need to find Lim (x->0) arcsin(2x)/arcsin(3x) by substitution

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To find the limit of arcsin(2x)/arcsin(3x) as x approaches 0, a substitution is made where arcsin(2x) is set to y, leading to the expression arcsin(sin(y)) for the numerator. The denominator becomes arcsin(3/2 sin(y), which complicates the calculation. The limit results in the indeterminate form 0/0, suggesting the use of L'Hôpital's rule for resolution. However, the discussion reveals that L'Hôpital's rule has not been covered in the current curriculum, leaving the user seeking alternative methods to solve the limit. The challenge lies in navigating the limit without the tools of L'Hôpital's rule.
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I need to find Lim (x->0) arcsin(2x)/arcsin(3x)

I can do a substitution
arcsin(2x) = y => 2x = sin(y)

and get arcsin(sin(y)) for the nominator, which is equal to y.
However, for the denominator i get arcsin(3/2 sin(y)) which I'm not sure what to do with.

Am I on the right path?
 
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Since arcsin(0)=0, you have a limit of the form 0/0...so why not use L'hopital's rule?
 


gabbagabbahey said:
Since arcsin(0)=0, you have a limit of the form 0/0...so why not use L'hopital's rule?

Well unfortunatelly i can't even tough i know it, because we haven't covered that part yet.
 

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