Solving lim(x->0)((cosx)^(-4/(x^2))) without L'Hopital's Rule lim(x->0)((cosx)^(-4/(x^2))) I would like to know how I can solve it without using De L'Hospital's rule. http://img803.imageshack.us/img803/7954/wolframalphalimxgt0cosx.png Thanks!
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You need to take logarithm of the limit such that:
`ln lim_(x->0) ((cosx)^(-4/(x^2))) => lim_(x->0) ln ((cosx)^(-4/(x^2)))`
You need to use the logarithmic identity such that:
`lim_(x->0) -4/(x^2) ln (cos x) = -oo/oo`
You may use l'Hospital's theorem such that:
`lim_(x->0) ((-4 ln (cos x))')/((x^2)') = lim_(x->0) (-4(-sin x)/cos x)/(2x)`
You need to substitute `tan x` for `sin x/cos x` such that:
`lim_(x->0) (-4(-sin x)/cos x)/(2x) = (4/2) lim_(x->0) tan x/x`
You need to use the remarcable limit such that:
`lim_(x->0) tan x/x = 1`
`lim_(x->0) -4/(x^2) ln (cos x) = (4/2)*1`
`lim_(x->0) -4/(x^2) ln (cos x) = 2`
You need to remember that `ln lim_(x->0) ((cosx)^(-4/(x^2))) = 2 =>lim_(x->0) ((cosx)^(-4/(x^2))) = e^2` .
Hence, evaluating the given limit using logarithmic identities yields `lim_(x->0) ((cosx)^(-4/(x^2))) = e^2.`
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