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Prove that lim(xrarr0) ((1+x)^(n) - 1)/(...

Prove that `lim_(xrarr0) ((1+x)^(n) - 1)/(x) = n`.

Text Solution

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`underset(x=0)"lim"((1-x)^(n)-1)/(x)`
`= underset(xrarr0)"lim"([1+nx+(n(n-1))/(2!)x^(2)+(n(n-1)(n-2))/(3!)x^(3)+"....."]-1)/(x)`
`= underset(xrarr0)"lim"[n+(n(n-1))/(2!)x+(n(n-1)(n-2))/(3!)x^(2)+"..."]`
`= n+0+0+"...."`
`= n`
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