Question

Hi there,

I'm currently stuck on a maths question.
Find the limit as x approaches 0 for (1+sinx)^cotx

I've put logs on both sides and am attempting to use hopitals rule but don't know where to proceed from here.

Answer 1

let's use an intuitive approach

You might recall that
lim (1 + 1/n)^n = e as n ---> ∞
or
lim (1 + n)^(1/n) = e , as n ---> 0


now we have (1+sinx)^(1/tanx)
now as x ---> 0 , sinx --->0 and 1/tanx ----> ∞
so we have
(1 + really small)^really large
which is e

test: on my calculator, I let x = .000001
and got
(1+sin(.000001)^(1/tan.000001)
= (1+.000001)^1000000
= 2.718280469
and e = 2.718281828 , not bad eh?