The sandwiich theorem?

Question

can someone help me with solve this

lim x-> 0

tan(5x)/tan(2x)
steps pls thx

Answer 1

since tan(n) = sin(n)/cos(n)
A = tan(5x)/tan(2x)
A = ( sin(5x)/cos(5x) ) / ( sin(2x)/cos(2x) )
A = ( sin(5x).cos(2x) ) / ( cos(5x).sin(2x) )
but when lim x -> 0 we can say sin(x) = x and cos(x) = 1
A = ( 5x.1 ) / ( 1.2x)
A = 5x/2x
A = 5/2

Answer 2

It is also true that tan(x) ~ x for small values of x. So one can say that as x->0, tan(x) -> x.

Therefore lim(x->0) tan(5x)/tan(2x) = 5x/2x = 5/2