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gillgill
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How do you start this problem?
lim x^3/(tan^3(2x))
x->0
lim x^3/(tan^3(2x))
x->0
gillgill said:how do you split tan^3(2x) into sin and cos?...sin^3(2x)/cos^3(2x) or sin(2x)^3/cos(2x)^3??
The limit of x^3/(tan^3(2x)) as x approaches 0 is 0. This can be solved using L'Hopital's rule or by recognizing that as x gets closer to 0, the value of tan(2x) also gets closer to 0, making the entire expression approach 0.
The limit is equal to 0 because as x approaches 0, the function becomes indeterminate (0/0) and can be simplified using L'Hopital's rule to get 0.
Yes, the limit can also be solved using the Maclaurin series expansion of tan(2x). This method involves rewriting the function as a polynomial and taking the limit as x approaches 0.
The limit is significant because it shows that the function has a vertical asymptote at x = 0. This means that as x gets closer to 0, the function gets closer and closer to a vertical line, creating a discontinuity in the graph.
This limit can be used in physics and engineering to calculate the rate of change of a function at a specific point. It can also be used in optimization problems to find the maximum or minimum value of a function.