Verify if limit of ln(1+x)/x is 1, x-->0

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In other words, we'll have to prove that:

lim [ln(1+x)]/x = 1

We'll re-write the function:

lim (1/x)*ln(1+x) = 1

We'll apply the power property of logarithms:

lim ln [(1+x)^(1/x)] = ln lim [(1+x)^(1/x)]

But, for x->0 lim [(1+x)^(1/x)] = e (remarcable limit)

lim ln [(1+x)^(1/x)] = ln e

We know that ln e = 1

**So, for x->0, lim ln [(1+x)^(1/x)] = 1.**

We have to verify that lim x-->0 [ ln(1+x)/x] = 1.

substituting x = 0, we get the indeterminate form 0/0, therefore we can use the l'Hopital's rule and substitute the numerator and denominator with their derivatives.

=> lim x-->0 [ ((1/(1+x))/1]

substituting x = 0

=> 1/(1 + 0)

=> 1

**This verifies that lim x-->0 [ ln(1+x)/x] = 1.**

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