高等数学试题1.计算lim(1/x2-cot2x)X→02lim(1-x)tanπx/2x→1

Question

高等数学试题
1.计算lim(1/x2-cot2x) X→0 2 lim(1-x)tanπx/2x→1

Answer 1

lim(x→0)(1/x^2-(cotx)^2)
=lim(x→0)[(cotx)^2-x^2]/x^2(cotx)^2
=lim(x→0)[(cosx)^2-x^2sinx^2]/[x^2cosx^2]
=lim(x→0)[-x^2+(1+x^2)cosx^2]/[x^2cosx^2]
=+∞
lim(x→1) (1-x)(tan(πx/2)
=lim(x→1)(1-x)/cot(πx/2)
=lim(x→1)(1-x)'/cot(πx/2)'
=lim(x→1) (-1)/[(π/2)/-(sinπx/2)^2]
=2/π