Question

Lim x tends to 0. Tanx/((1-cosx)^2)^1/3)

Answer 1

Lim(x→0) Tanx/{(1 - cosx)²}^1/3

first of all we check form of limit ,
put x = 0 we see limit is 0/0 form .
hence, we can use , L- Hospital rule .
but it's tough so, we use concept ,
Lim(x→0) sinx/x = 1
Lim(x→0) tanx/x = 1

now , starting ;

Lim(x→0) {tanx/x}x/{(2sin²x/2)²}^1/3
[ use,(1 - cosx) = 2sin²x/2 ]
Lim(x→0) x/{x⁴/16)⅓
1/16⅓ ×Lim(x→0) x/x^4/3
you can see here, denominator have more power then numerator .e.g 4/3 > 1
so, if we put x = 0 limit gain, ∞ hence,
limit doesn't exist .