Question

compute limit [x->0] (sqrt(cos(x)) - (cos(x))^(1/3))/(sin(x)^2)

Answer 1

\[\lim_{x \rightarrow 0} \frac{ \sqrt{\cos(x)} - \sqrt[3]{\cos(x)} }{ \sin^2(x) }\]

Answer 2

Are you allowed to use L'Hopital's rule?

Answer 3

no :(

Answer 4

That sucks. I'm trying to figure it out without it then haha. Give me a second, I am trying to multiply it by [cos^(1/2)x+cos^(1/3)x]/[cos^(1/2)x+cos^(1/3)x] right now.

Answer 5

Actually before that, maybe you should try splitting the limit up into two parts and solving those individually.

Answer 6

What sort of tricks are you allowed to use if not L'H rule?

Answer 7

Every other tricks are allowed ;)

Answer 8

Done.
Here: http://math.stackexchange.com/questions/628137/compute-limit-no-lhospital-rule?answertab=active#tab-top

Answer 9

\[\lim_{x \rightarrow 0}\frac{ \sqrt{\cos x}-\left( \cos x \right)^{\frac{ 1 }{ 3 }} }{\sin ^{2}x }\]
\[=\lim_{x \rightarrow 0}\frac{ \sqrt{\cos x}-1+1-\left( \cos x \right)^{\frac{ 1 }{ 3 }} }{ \sin ^{2}x }\]
\[=\lim_{x \rightarrow 0}\frac{ \sqrt{\cos x}-1 }{\sin ^{2}x }+\lim_{x \rightarrow 0}\frac{ 1-\left( \cos x \right)^{\frac{ 1 }{ 3 }} }{ \sin ^{2 }x}=L1+L2\]
for L1 multiply the numerator and denominator by \[\sqrt{\cos x}+1 and solve\]

Answer 10

\[1+\left( \cos x \right)^{\frac{ 1 }{ 3 }}+\left( \cos x \right)^{\frac{ 2 }{ 3 }}\]

Answer 11

multiply the numerator and denominator of L2 as written above.