finding the limits?

Question

Find the limit for the given function.
lim_(x -> 0) (sin 2 x)/ x

Find the limit for the given function.
lim_(x->0) (sin(5 x))/(sin(6 x))

Find the limit for the given function.
lim_(t->0) (sin)^2(5 t))/t^2

Answer 1

lim_(x -> 0) (sin 2 x)/ x
= lim_(x -> 0) (sin 2 x)/ (2x) * 2
= lim_(2x -> 0) (sin 2 x)/ (2x) * 2
= 1* 2 = 2

lim_(x->0) (sin(5 x))/(sin(6 x))
= lim_(x->0) (sin(5 x)/(5x))/(sin(6 x)/6x) * (5x/6x)
= lim_(5x->0) (sin(5 x)/(5x))/ lim_(6x->0) (sin(6 x)/6x) * (5/6)
= 1/1 *(5/6)
= 5/6

lim_(t->0) (sin)^2(5 t))/t^2
= lim_(t->0) (sin(5 t)/t)^2
= lim_(t->0) (sin(5 t)/5t)^2 * 5^2
= lim_(5t->0) (sin(5 t)/5t)^2 * 5^2
= (lim_(5t->0) (sin(5 t)/5t))^2 * 5^2
= 1^2 * 25
= 25

Answer 2

Look up L'Hospital's Rule in your text book