1. Prove that limx→∞aex+be−xcex+de−x=ac[2<e<3,c≠0]
Solution:
limx→∞aex+be−xcex+de−x
=limx→∞a+be−2xc+de−2x
=a+be−∞c+de∞
=ac Proved
2. Prove that limx→−∞aex+be−xcex+de−x=bd[2<e<3,d≠0]
Solution:
limx→−∞aex+be−xcex+de−x
=limx→−∞ae2x+bce2x+d
ae−∞+bce−∞+d
=bd Proved
4. Prove that limx→14x−4x−1=8loge2
Solution:
limx→14x−4x−1
=4limx−1→04(4x−1−1)x−1
=4limx−1→0(4x−1−1)x−1
=4loge4
=8loge2 Proved
5. Prove that limx→2log(2x−3)2(x−2)=1
Solution;
limx→2log(2x−3)2(x−2)
=lim2x→4log(2x−4+1)2x−4)
=lim2x−4→0log(2x−4+1)2x−4)
=1 Proved
6. Show that limx→0(1+2x)1x=e2
Solution:
limx→0(1+2x)1x
=limx→0(1+2x)1⋅22x
=[limx→0(1+2x)12x]2
=e2 proved [Since limx→0(1+x)1x=e]
7. Show that limx→0(1−3x)3x=e−9]
Solution:
limx→0(1−3x)3x
=limx→0(1−3x)−13x⋅(−9)
=[lim−3x→0(1−3x)−13x]−9
=e−9 Proved [Since limx→0(1+x)1x=e]
8. Show that limx→1x11−x=e−1
Solution:
limx→1x11−x
=limx→1(1+x−1)11−x⋅(1)
=[limx−1→0(1+x−1)11−x]−1
=e−1 Proved [Since limx→0(1+x)1x=e]
9. Show That limx→0(1+x)x+2x=e8
Solution:
limx→0(1+x)x+2x
=limx→0(1+x)14x⋅(4x+8)
=[lim4x→0(1+x)14x]limx→0(4x+8)
=e8 Proved [Since limx→0(1+x)1x=e]
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