SOLUTION: Please evaluate :
Lim x tends to zero, ((tanx)/x)^(1/x^2) with step by step solution. The answer is = e^(1/3) as per book.
Regard
Khoka123
Algebra.Com
Question 890090: Please evaluate :
Lim x tends to zero, ((tanx)/x)^(1/x^2) with step by step solution. The answer is = e^(1/3) as per book.
Regard
Khoka123
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
}{x}\right)^{\frac{1}{x^2}})
Exponentiate:
}{x}\right)}\ =\ \lim_{x\rightarrow 0}\ e^{\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2})
Using the fact that 
is continuous at 
and if )
is continuous around )
and )
is continuous around 
, then )\ =\ f\left( \lim_{x\rightarrow a}\,g(x)\right))
}{x}\right)}{x^2}}\ =\ e^{\lim_{x\right 0}\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}})
Use a Taylor Series expansion at to compute }{x}\right)}{x^2})
\ =\ \frac{1}{3})
Hence,
}{x}\right)}{x^2}}\ =\ e^{\frac{1}{3}})
John
My calculator said it, I believe it, that settles it
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