(4)
lim(x->1) (x^n-1)/(x-1)
=lim(x->1) [x^(n-1)+x^(n-2)+...+x+1]
=1+1+...+1
=n
(2)
f(x) =xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
case 1: x<-1
f(x)
=xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
=xlim(n->∞) [1/x^(2n)-1)]/[1/x^(2n)+1 ]
=x(0-1)/(0+1)
=-x
case 2: x=-1
f(x) =xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
f(-1) = x( 1-1)/(1+1) =0
case 3: -1<x<1
f(x)
=xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
=x(1-0)/(1+0)
=x
case 4: x=1
f(x) =xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
f(1) = x( 1-1)/(1+1) =0
case 5 : x>1
f(x)
=xlim(n->∞) [1-x^(2n)]/[1+x^(2n) ]
=xlim(n->∞) [1/x^(2n)-1)]/[1/x^(2n)+1 ]
=x(0-1)/(0+1)
=-x
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