SOLUTION: Please help me solve this equation: Evaluate the following limits or explain why they do not exist. a. {{{ lim(x->0, cotx(1-cosx) ) }}} b. {{{ lim (x->0, (sinx)^tanx ) }}}

Algebra ->  Test -> SOLUTION: Please help me solve this equation: Evaluate the following limits or explain why they do not exist. a. {{{ lim(x->0, cotx(1-cosx) ) }}} b. {{{ lim (x->0, (sinx)^tanx ) }}}      Log On


   

Question 363622: Please help me solve this equation:
Evaluate the following limits or explain why they do not exist.
a. +lim%28x-%3E0%2C+cotx%281-cosx%29+%29+
b. +lim+%28x-%3E0%2C+%28sinx%29%5Etanx+%29+
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
a. +lim%28x-%3E0%2C+cotx%281-cosx%29+%29++=+lim%28x-%3E0%2C+cosx%281-cosx%29%2Fsinx+%29
=lim%28x-%3E0%2C+cosx%29*lim%28x-%3E0%2C+%281-cosx%29%2Fsinx+%29 = 1*0 = 0. The second limit was obtained by applying LHopital's rule.
b. Let y=+%28sinx%29%5Etanx. then lny+=+tanx%2Alnsinx+=+lnsinx%2Fcotx. The limit as x approaches 0 has the form infinity%2Finfinity, so apply the LH rule. The new expression becomes %28cosx%2Fsinx%29%2F%28-csc%5E2x%29. Simplifying this, we get cosx%2F%28-cscx%29+=+-sinxcosx. As x goes to 0, -sinx*cosx approaches 0. Hence, lny approaches 0 as x approaches 0, and thus y approaches 1.