Question

How do you find the limit of `(tan^3 (2x))/ x^3` as x approaches 0?

Answer 1

`8`

Explanation:

`lim_(x->0) tan^3(2x)/x^3=8lim_(x->0) tan^3(2x)/(8x^3)=8lim_(x->0) tan^3(2x)/(2x)^3=`

`=8lim_(x->0) (tan(2x)/(2x))^3=8(lim_(x->0) (tan2x)/(2x))^3=`

`=8(lim_(x->0) (sin2x)/(2x) 1/cos(2x))^3 =`
`=8(lim_(x->0) (sin2x)/(2x) lim_(x->0) 1/cos(2x))^3 = A`

`2x=t => x->0 =>t->0`

`A=8(lim_(t->0) (sint)/(t) lim_(t->0) 1/cost)^3 = 8 * (1*1)^3 = 8`