Learn How To Solve Limit Problems With L’Hopital’S Rule And Natural Logarithm Approach

the limit as n approaches zero of (1+n)¹/ⁿ is

lim n→0 (1+n)¹/ⁿ = e.

To solve this limit problem, we can use L’Hopital’s rule or the natural logarithm approach.

Using L’Hopital’s rule:

Taking the natural logarithm of the expression and using logarithmic properties, we get

lim n→0 ln(1+n) / n

Now applying L’Hopital’s rule by taking the derivative of the numerator and denominator with respect to n,

= lim n→0 (1 / (1+n)) / 1

= (1 / (1+0)) / 1

= 1

Therefore, lim n→0 (1+n)¹/ⁿ = e¹ = e.

Using the natural logarithm approach:

We can use the fact that

lim n→0 (1+n)¹/ⁿ = lim n→0 e^(ln(1+n) / n)

Now, applying the limits property of exponentials, we get

= e^lim n→0 ln(1+n) / n

And again, using L’Hopital’s rule,

= e^(lim n→0 1 / (1+n))

= e^(1/1)

= e

Therefore, lim n→0 (1+n)¹/ⁿ = e.

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