Question

How do you compute the limit of `tanx/x` as `x->0`?

Answer 1

Rewrite and use `lim_(xrarr0) sinx/x = 1` and cosine is continuous at `0`.

Explanation:

`lim_(xrarr0) tanx/x = lim_(xrarr0) (sinx/cosx)/x`

`= lim_(xrarr0) sinx/(xcosx)`

`= lim_(xrarr0) (sinx/x*1/cosx)`

`= lim_(xrarr0) (sinx/x) *lim_(xrarr0)(1/cosx)` (provided that both limits exist)

`= (1)(1/1) = 1`