lim_x→ 0(1+x)^n 1/x or lim_x→ 0(1+x)^n (1)^n/(1+x) (1) (lim x^n a^nx a=nx^n 1) ⇒ n(1+x)^n 1 ⇒ n(1+0)^n 1 =n(1)^n 1=n ...

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Algebra of Limits

lim x→ 0 1+x ...

Question

${lim}_{x \to 0} \frac{{(1 + x)}^{n} - 1}{x}$

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Solution

$lim x \to 0 \frac{(1 + x)^{n} - 1}{x}$
or $lim x \to 0 \frac{(1 + x)^{n} - (1)^{n}}{(1 + x) - (1)}$
$(l i m \frac{x^{n} - a^{n}}{x - a} = n x^{n - 1})$
$\Rightarrow n (1 + x)^{n - 1}$
$\Rightarrow n (1 + 0)^{n - 1}$
$= n (1)^{n - 1} = n$

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