Part 2:
limx→0(sin(x)x)cot(x)= ?(sin(x)x)cot(x)=eln(sin(x)x)⋅cot(x)=eln(1+sin(x)−xx)⋅cot(x)ln(1+sin(x)−xx)⋅cot(x)=[(−x23!+x45!−x67!+x89!−+⋯)1−12(−x23!+x45!−x67!+x89!−+⋯)2+13(−x23!+x45!−x67!+x89!−+⋯)3−14(−x23!+x45!−x67!+x89!−+⋯)4]⋅(1x−13x−145x3−2945x5−14725x7−⋯)
ln(1+sin(x)−xx)⋅cot(x)=[(−x26+x4120−x65040+x840320−+⋯)−12(x436−x6360+41x8302400−+⋯)+13(−x6216+x81440−+⋯)−14(x81296−+⋯)]⋅(1x−13x−145x3−2945x5−14725x7−⋯)
until x7:
ln(1+sin(x)−xx)⋅cot(x)=×(1x):−x6+x3120−x55040+x740320−+⋯−x372+x5720−41x7604800+−⋯−x5648+x74320−+⋯−x75184+−⋯×(−1x3):+x318−x5360+x715120−+⋯+x5216−x72160+−⋯+x71944−+⋯×(−1x345):+x5270−x75400−+⋯+x73240−+⋯×(−2x5945):+x72835−+⋯
ln(1+sin(x)−xx)⋅cot(x)=−x6+x3(1120−172+118)+x5(−15040+1720−1648−1360+1216+1270)+x7(140320−41604800+14320−15184+115120−12160+11944−15400+13240+12835)+⋯ ln(1+sin(x)−xx)⋅cot(x)=−x6+x320+59x511340+401x7680400+⋯ limx→0(sin(x)x)cot(x)=limx→0eln(sin(x)x)⋅cot(x)=limx→0eln(1+sin(x)−xx)⋅cot(x)=limx→0e−x6+x320+59x511340+401x7680400+⋯=e0 limx→0(sin(x)x)cot(x)=1
ready.
