Question

find the limit algebraically. Use L'Hospital's Rule where appropriate. If there is a moare elementary method, consider using it. If L'Hospital's Rule doesn't apply, explain why

lim -> 0 (cot(x)-(1/x))

show work please!!!

Answer 1

lim -> 0 (cot(x)-(1/x)) , x---> 0
= sec^2 x + 1/x^2 by L'Hospital's Rule

but that is just as bad

try experimentally ... using a Calculator
let x = .0001
I get -.0000333

let x = .000001
my calculator says 0

http://www.wolframalpha.com/input/?i=limit+cot%28x%29+-+1%2Fx+%2C+x+%3D+0

the red dot shows the value of the limit

Answer 2

Hmmm. we all know that
tan x -> x as x->0
so, cot x -> 1/x as x->0

so the limit ought to be zero.