# Trigo Limit Type 0/0

1. Jul 15, 2011

### Deathfish

1. The problem statement, all variables and given/known data
Find limit of (sin x - x)/(x - tan x) as x approaches zero

2. Relevant equations
Type 0/0 , use L'Hopital's rule, differentiate.

3. The attempt at a solution
Every time I apply the rule it gets more complicated

(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)
(cos x)/(2 Sec^4 [x] - 4 Sec^2 [x] Tan^2 [x])

2. Jul 15, 2011

### micromass

Staff Emeritus
Stop here, write sec and tan in terms of sin and cos. Eliminate sin from numerator en denominator.

3. Jul 15, 2011

### Deathfish

Ok usually how do you decide when to stop differentiation and use alternative method?

4. Jul 15, 2011

### micromass

Staff Emeritus
The idea is to simplify the formula enough after each differentiation. That way you can see that you need to stop differentiation. Here, you need to write everything in sin and cos. Then you'll see after the second differentiation that you can cancel thingies.