Evaluate without L'Hopital: lim x → 3 ( x − 3 ) csc ⁡ π xI find the indeterminate form of 0 or 0 0 . The latter tells me that L'Hopital's is an option, but since we haven't seen derivatives yet I'm not allowed to used it.Previously I already tried swapping the csc ⁡ π x for 1 sin ⁡ π x but when doing this I can't seem to get rid of the sinus. I also believe that since the limit goes to 3 and not to 0, the lim x → 0 sin ⁡ a x a x = 1 rule is not an option either.I tried making the sin ⁡ ( π 2 ) so that =1, but without any success. Can anyone give me a hint?

Question

Evaluate without L&#039;Hopital:       lim          x      →      3        (  x  −  3  )  csc  ⁡  π  xI find the indeterminate form of 0 or       0    0  . The latter tells me that L&#039;Hopital&#039;s is an option, but since we haven&#039;t seen derivatives yet I&#039;m not allowed to used it.Previously I already tried swapping the   csc  ⁡  π  x for       1          sin      ⁡      π      x       but when doing this I can&#039;t seem to get rid of the sinus. I also believe that since the limit goes to 3 and not to 0, the       lim          x      →      0                  sin      ⁡      a      x              a      x        =  1 rule is not an option either.I tried making the   sin  ⁡  (      π    2    ) so that =1, but without any success. Can anyone give me a hint?

garcialdariamcy4q · Accepted Answer

Hint:   sin  ⁡  (  x  )  =  −  sin  ⁡  (  x  +  3  π  ). Try a substitution such that the limit you mentioned is an option!

hisyhauttaq84w · Accepted Answer

By a change of variable,      lim          x      →      3                  x      −      3              sin      ⁡      π      x        =      lim          y      →      0            y          sin      ⁡      (      π      y      +      3      π      )        =  −      1    π        lim          y      →      0                  π      y              sin      ⁡      (      π      y      )        .

Evaluate without L'Hopital: <munder> <mo movablelimits="true" form="prefix">lim <mrow cla

Answered question

Answer & Explanation

New Questions in Differential Calculus