次の極限を求めよ lim x→0 1+(1/sinx) lim x→1 (x^3-2x^2+2x+1)/(x^2-3x+2 ) ...

lim x→0 1+(1/sinx)
=1±∞
=±∞

lim x→1 (x^3-2x^2+2x+1)/(x^2-3x+2 )
=(1-2+2+1)/(1-3+2)
=∞

lim x→-1 (1-|x|)/(|1+x|)
=lim t→0 (1-|t-1|)/|t|=lim t→0 (t)/|t|
=(1,-1)

lim x→0 (e^x-e^-x)/sin2x
=lim x→0 (e^2x-1)/(e^x)sin2x
=lim x→0 (e^2x-1)/(e^x)*1/(2x)*(2x/sin2x)
=lim x→0 (e^2x-1)/(2x)*1/(e^x)*(2x/sin2x)
=lim x→0 (2e^2x)/(2)*1/(e^x)*(2x/sin2x)
=1*1*1
=1
(途中微分ではなくロピタルを使用)

lim x→+0 √xsin(1/x)
=lim t→+0 √1/tsin(t)
=lim t→+0 sint/√t
=lim t→+0 (sint/t)*√t=1*0
=0

arcsinx=y
x=siny
x→0,y→0
lim x→0 (arcsinx/tanx)
=lim x→0,y→0 (ycosx/sinx)
=lim x→0,y→0 (ycosx)*(x/sinx)*1/x
=lim x→0,y→0 (ycosx)*(x/sinx)*1/siny
=lim x→0,y→0 (y/siny)cosx)*(x/sinx)
=1*1*1
=1