Use l'Hopital's rule to evaluate the one-sided limits. Support your answer graphically. The limit as x approaches 0 from the left of sinx/x^3.

Question

Use l'Hopital's rule to evaluate the one-sided limits. Support your answer graphically.  The limit as x approaches 0 from the left of sinx/x^3.

hartnn · Answer

you know what L'Hopital's rule is ?

anonymous · Answer

Yes, how would I use it in this case?

hartnn · Answer

whats the derivative of numerator ?
whats the derivative of denominator ?

anonymous · Answer

Cosx/ 3x^2

hartnn · Answer

correct! 
now tell me whether you can use L'Hopital again ??

anonymous · Answer

I cannot now b/c its it's some number over 0. I just don't know which infinity it is and why.

hartnn · Answer

ok! the hint is in 
x approaches 0 from the "left"
you know what that means ? 
x>0 or x <0 ?

anonymous · Answer

X<0?

hartnn · Answer

correct!
since x< 0 
the denominator , obviously is positive, right ? (x^2 always positive)
what about the numerator ?

anonymous · Answer

The numerator is negative

hartnn · Answer

really ? 
when is cos x negative and positive ?

anonymous · Answer

Oh I see, cosx is positive on the right side of the unit circle, meaning that the answer is positive infinity. Thanks so much!

hartnn · Answer

absolutely correct! 
cos is positive in 1st and 4th quadrant, so numerator also positive :)