Limits Question (dividing by zero?)

Question

Calculate: lim x->0 (sin4x²)/(x²)

 

I've seen an explanation of the problem and it says the answer is 4. I just don't understand what you're supposed to do after u-substitution. More specifically, I don't understand how lim u->0 (sin u)/u=1. Isn't 0/0 undefined? I feel like I'm forgetting a key concept here...

 

Thank you in advance!

 

 

Answer 1

There are multiple ways to solve this. We could use L'Hopital's rule for limit in which we take a derivative of the numerator and denominator until we get an answer other than 0/0. 

 

lim x-->0 (sin 4x^2)/(x^2)

 

lim x-->0 8x(cos 4x^2)/2x =

 

lim x-->0 4*cos (4x^2) = 4