Use the fundamental limit `lim_(t->0)(1+t)^(1/t)=e` to find `lim_(x->0)(1+x/5)^(1/x).`      

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Let `t=x/5,` so that `1/t=5/x.` Then as `x->0,` so does `t` and we get

`lim_(x->0)(1+x/5)^(5/x)=lim_(t->0)(1+t)^(1/t)=e.` The left side isn't quite what we're after, but ` `

 

`lim_(x->0)(1+x/5)^(1/x)=lim_(x->0)(1+x/5)^(5/x*1/5)=(lim_(x->0)(1+x/5)^(5/x))^(1/5)=e^(1/5).`

The limit is `e^(1/5).`

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