Starting a Limit Problem with x^3/(tan^3(2x)) as x Approaches 0

[gillgill]

How do you start this problem?
lim x^3/(tan^3(2x))
x->0

 

[Data]

Use

\tan{x} = \frac{\sin{x}}{\cos{x}}, \; \mbox{and} \ \lim_{x \rightarrow 0} \frac{\sin{\alpha x}}{\alpha x} = 1

 

[gillgill]

how do you split tan^3(2x) into sin and cos?...sin^3(2x)/cos^3(2x) or sin(2x)^3/cos(2x)^3??

 

[xanthym]

how do you split tan^3(2x) into sin and cos?...sin^3(2x)/cos^3(2x) or sin(2x)^3/cos(2x)^3??

1): \ \ \ \ \tan^{3}(2x) \ = \ \frac {\sin^{3}(2x)} {\cos^{3}(2x)}

2): \ \ \ \ \Longrightarrow \ \ \frac {x^{3}} { \tan^{3}(2x)} \ = \ \frac {\cos^{3}(2x)} { \frac {\sin^{3}(2x)} {x^{3}} } \ = \ \frac {\cos^{3}(2x)} { \frac {\sin^{3}(2x)} {(1/8) \cdot (2x)^{3}} } \ = \ \left( \frac{1}{8} \right) \cdot \left ( \frac {\cos^{3}(2x)} { \left ( \frac {\sin(2x)} {(2x)} \right )^{3} } \right )

Now use info provided by Data in MSG #2 to evaluate required Limit.


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[gillgill]

thanks...^^