1樓：匿名使用者
第一張紙
(1)lim(x->0) (a^x -b^x) /x
=lim(x->0) [(1+xlna) -(1+xlnb)] /x
=lna -lnb
(2)lim(x->0) [cosx -√(1+x) ]/x
=lim(x->0) /x
=-1/2
(3)x->0
e^x -e^(-x) -2x
~ ( 1+x +(1/2)x^2 + (1/6)x^3) -( 1-x +(1/2)x^2 - (1/6)x^3) -2x
~(1/3)x^3
------
sinx ~ x- (1/6)x^3
x-sinx ~ (1/6)x^3
-------
lim(x->0) [e^x-e^(-x) -2x ]/(x-sinx)
=lim(x->0) (1/3)x^3/[(1/6)x^3]
=2(4)
a^x ~ 1+ x(lna) +(1/2) x^2. (lna)^2 + (1/6)x^3 .(lna)^3
a^(sinx)
~ a^(x - (1/6)x^3)
~ 1+ (x-(1/6)x^3)(lna) +(1/2) (x-(1/6)x^3)^2. (lna)^2 + (1/6)(x - (1/6)x^3)^3 .(lna)^3
~ 1+ x(lna) +(1/2)x^2.(lna)^2 +[(-1/6)lna +(1/6)(lna)^3 ]x^3
a^x - a^(sinx)~ -(1/6)(lna).x^3
----------
lim(x->0) [a^x-a^(sinx)]/(sinx)^3
=lim(x->0) -(1/6)(lna)x^3/x^3
=-(1/6)(lna)
第2張紙
(1)lim(x->+∞) lnx/x^n (∞/∞)
=lim(x->+∞) (1/x)/[nx^(n-1)]
=lim(x->+∞) 1/[nx^n]
=0(2)
lim(x->0+) lnsinx/ln(sin√x) (0/0)
=lim(x->0+) [cosx/sinx]/[ (cos√x/sin√x) .(1/(2√x)) ]
=lim(x->0+) 2√x tan√x / tanx
=lim(x->0+) 2x /x
=2(3)
lim(x->+∞) e^x/lnx (∞/∞)
=lim(x->+∞) e^x/(1/x)
=lim(x->+∞) x.e^x
->∞(4)lim(x->+∞) (e^x-x)/(e^x +x)
=lim(x->+∞) [1 - 2x/(e^x +x)]
=1 - lim(x->+∞) 2x/(e^x +x)]
= 1- 0
=1(5)
lety=1/x
lim(x->+∞) x[e^(1/x) -1]
=lim(y->0) (e^y -1)/y
=lim(y->0) y/y
=1(6)
lety=x-1
y->0
(1+y)ln(y+1) ~ y+y^2
(1+y)ln(y+1) -y ~ y^2
------
lim(x->1) [ x/(x-1) - 1/lnx )
=lim(y->0) [ (1+y)/y - 1/ln(y+1) ]
=lim(y->0) [ (1+y)ln(y+1) -y ] /[y.ln(y+1) ]
=lim(y->0) y^2/y^2
=1(7)
lety= x-π/2
lim(x->π/2) ( secx -tanx)
=lim(x->π/2) ( 1/cosx -tanx)
=lim(y->0) ( 1/siny -coty)
=lim(y->0) ( 1-cosy)/siny
=lim(y->0) ( 1/2)y^2)/y
=0(8)
l =lim(x->0+) (sinx)^[2/(1+lnx)]
lnl=lim(x->0+) 2ln(sinx)/(1+lnx) (∞/∞)
=lim(x->0+) 2cotx/(1/x)
=2lim(x->0+) x/tanx
=2lim(x->0+) 2ln(sinx)/(1+lnx) =l = e^2
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