Question

Prove that t approaches to 0 lim(1+t)1/t=e

Answer 1

Lim( t→∞) {(1 + t)^1/t }

put t = 0, we are seeing limit in the form of 1^∞ ,

now, (1 + t)^1/t = e^ln( 1 + t)/t
put this in Limit in place of (1 + t)^1/t

Lim( t →0) {e^ln( 1 + t)/t }
use expansion of log(1 +t)
log( 1 + t) = t -t²/2 + t³/3 -........∞

=Lim( t→∞) { e^{t -t²/2 +t³/3 -......∞)/t }
=Lim(t→∞){ e^{ 1 -t/2 +t³/3 -.......∞)}

put t = 0 then,
= e^{ 1 - 0 + 0...∞} = e

hence,
Lim(t→0) { (1 + t)^1/t } = e