Question

求第5题和第7题的极限

Answer 1

(5)
x->0+
√(1+ √x) ~ 1+ (1/2)√x
√(1+ √x) -1 ~ (1/2)√x

sin√x ~√x

lim(x->0+) [√(1+ √x) -1]/sin√x
=lim(x->0+) (1/2)√x /√x
=1/2

(7)
x->0
tanx~ x +(1/3)x^3
x-tanx ~ -(1/3)x^3

e^x ~ 1+ x + x^2/2 + x^3/6

e^(tanx)
~ e^(x + (1/3)x^3 )
~ 1+ (x + (1/3)x^3 ) + (1/2)[x + (1/3)x^3 ]^2 +(1/6)[(x + (1/3)x^3 )]^3
~ 1+ x +(1/2)x^2 + (1/2)x^3

e^x - e^(tanx)
~[1+ x + x^2/2 + x^3/6] -[1+x +(1/2)x^2 + (1/2)x^3]
= -(1/3)x^3

lim(x->0) [ e^x -e^(tanx) ]/ (x-tanx)
=lim(x->0) [ -(1/3)x^3 ]/ (-(1/3)x^3)
=1