l'Hôpital's rule.

Question

lim x approaches 0 x / (3+sin(x)

why is the answer zero instead of one

the derivative is 1 / cos(x) and if i plug zero into cos i get one so 1/1 is equal to 1

Answer 1

As x/(3+sin(x)) is not in an indeterminate form (ex. 0/0, ∞/∞, 0·∞, 1^∞), this doesn't make l'Hopital's rule work. This means we can just directly plug in the limit:

lim_x->0 x/(3+sin(x))

= 0/(3+sin(0))

= 0/(3+0)

= 0/3

= 0

Hence, lim_x->0 x/(3+sin(x)) = 0 by direct substitution. Hope this helped!

Answer 2

x/(3 + sinx)

plug in 0 and it's 0/3 = 0

no need for L'Hopital's rule

but if you tried to use it, then

derivatives of numerator and denominator gives you

1/cosx, plug in 0 for x does give you 1/1 = 1

but if directly plugging the x limit into f(x) isn't an indeterminate form, don't try to use L'Hopital's rule

but you're right if you used it twice you'd get: 0/-sinx = 0/-0 = an undefined indeterminant form

so doing it twice won't work either

do it 3 times and you get 0/-cosx = 0/-cos0 = 0/-1 = 0