SOLUTION: Evaluate the following limits or show they do not exist: a) lim x→0 {{{ (sin(x))^2/(x(1-cos(x))) }}} b) lim x→0 {{{ (sin(ax))/(sin(bx)) }}} where a,b are nonzero

Algebra ->  Expressions -> SOLUTION: Evaluate the following limits or show they do not exist: a) lim x→0 {{{ (sin(x))^2/(x(1-cos(x))) }}} b) lim x→0 {{{ (sin(ax))/(sin(bx)) }}} where a,b are nonzero       Log On


   

Question 365859: Evaluate the following limits or show they do not exist:
a) lim x→0 +%28sin%28x%29%29%5E2%2F%28x%281-cos%28x%29%29%29++
b) lim x→0 ++%28sin%28ax%29%29%2F%28sin%28bx%29%29+ where a,b are nonzero real numbers
c) lim x→1 +%28x%5E3-1%29%2Fabs%28x%5E3-1%29+
d) lim x→2 +%281-2%2Fx%29%2F%281-4%2Fx%5E2%29+
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
b) Answer is a/b, by L'Hopital's rule.
c) Limit does not exist: left and right hand limits are not the same. Right hand limit is 1, while the left hand limit is -1.
d) +%281-2%2Fx%29%2F%281-4%2Fx%5E2%29+=+1%2F%281%2B2%2Fx%29+. Therefore the limit is +1%2F%281%2B2%2F2%29+=+1%2F%281%2B1%29+=+1%2F2+