1. Prove that limxaex+bexcex+dex=ac[2<e<3,c0]

Solution:

limxaex+bexcex+dex

=limxa+be2xc+de2x

=a+bec+de

=ac  Proved






2. Prove that limxaex+bexcex+dex=bd[2<e<3,d0]

Solution:

limxaex+bexcex+dex

=limxae2x+bce2x+d

ae+bce+d

=bd Proved






4. Prove that limx14x4x1=8loge2

Solution:

limx14x4x1 

=4limx104(4x11)x1

=4limx10(4x11)x1

=4loge4

=8loge2 Proved







5. Prove that limx2log(2x3)2(x2)=1

Solution;

limx2log(2x3)2(x2)

=lim2x4log(2x4+1)2x4)

=lim2x40log(2x4+1)2x4)

=1 Proved







6. Show that limx0(1+2x)1x=e2

Solution:

limx0(1+2x)1x

=limx0(1+2x)122x

=[limx0(1+2x)12x]2

=e2   proved            [Since limx0(1+x)1x=e]










7. Show that limx0(13x)3x=e9]

Solution:

limx0(13x)3x

=limx0(13x)13x(9)

=[lim3x0(13x)13x]9

=e9  Proved        [Since limx0(1+x)1x=e]









8. Show that limx1x11x=e1

Solution:

limx1x11x

=limx1(1+x1)11x(1)

=[limx10(1+x1)11x]1

=e1  Proved        [Since limx0(1+x)1x=e]








9. Show That limx0(1+x)x+2x=e8

Solution:

limx0(1+x)x+2x

=limx0(1+x)14x(4x+8)

=[lim4x0(1+x)14x]limx0(4x+8)

=e8    Proved        [Since limx0(1+x)1x=e]