Lim x>o (sec(4x))^(3/x)

[Sophie]

Hello my answer book says the following is wrong, however I belive it is write. Could somone either confirm I am correct and the answer book is wrong or show me where I have gone wrong (maybe I should have used the 3!). Thanks

Find:

Lim x > o ((sec(4x))^(3/x))

lny = (3/x) (ln(sec4x))

3 Lim x>0 ((ln sec 4x)/4)

This limit has the form 0/0

H= ((4secxtanx/sec4x)/1) = (4secxtanx/sec4x)

= (4(1)(0))/(1) = 0

y = e^0 = 1

therefore: Lim x > o ((sec(4x))^(3/x)) = 1

Thanks Sophie

 

[royhaas]

I took another route and came to the conclusion that you are correct.

 

[morson]

Do you not have a calculator? Can't you just plug in values of x very close to 0 from the positive and negative side and see if there is a unique limit?

The exponent surpasses extremely large numbers and approaches infinity, while the base tends to 1.