anonymous
  • anonymous
lim t->0 t^3 /tan^3(2t) ????
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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anonymous
  • anonymous
Near 0z we have tan x behave like x so our limit behaves like t^3/(8t^3) = 1/8
anonymous
  • anonymous
as \(t\to0\) we observe \(\tan 2t\approx2t\) and therefore \(\tan^32t\approx(2t)^3=8t^3\)
primeralph
  • primeralph
|dw:1371544705566:dw|

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anonymous
  • anonymous
our limit then behaves like \(\dfrac{t^3}{8t^3}=\dfrac18\) as \(t\to0\)
primeralph
  • primeralph
|dw:1371544716644:dw|
primeralph
  • primeralph
@oldrin.bataku Hefty assumption. Even though it's true, you might want to give a quicker form of convergence to 2t.
anonymous
  • anonymous
@primeralph it follows trivially from the power series expansions....$$\sin x\approx x\\\cos x\approx 1\\\tan x\approx x$$
primeralph
  • primeralph
@oldrin.bataku I know that. I said sinx tends to 1quicker than tanx; hence it is more appropriate for parts that have powers greater than 1.
anonymous
  • anonymous
Thank you primeralph and oldrin.bataku for your genuine help!!!

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