L'Hospital's Rule?

Question

how would you use it for this

lim x-->0 (arctan(5x))/(sin(7x))

Answer 1

d (arctan(5x)/dx = 1/(1+25x^2)
d(sin(7x)/dx=-cos (7x)/7

X -->0 1/(1+25x^2)/-cos (7x)/7 =1/1/7=7
X -->0 f(x) =7
Ithink i may have made a mistake you better check it on a graphic calculator to be sure

Answer 2

lim x-->0 (arctan(5x))/ lim x-->0 (sin(7x))

=0/0

so use L'Hospital's Rule

lim x-->0 (cos(5x)/-sin(5x))5 / lim x-->0 (cos(7x)7

=5/7

Answer 3

(5/1+25x^2)/7cos7x

limit is 5/7 for x=0