Question

evaluate the limit of ( 1 + ¹/ₓ)ˣ as x approaches infinity​

Answer 1

$\\e = \lim_{x \to \infty } {(1 + \frac{1}{x} )}^{x}$

$\\ =\lim_{x \to \infin}({e}^{ \ln({(1 + \frac{1}{x} )}^{x} ) }) = {e}^{ \lim_{x \to \infty } \frac{ \ln (1 + \frac{1}{x} )}{ \frac{1}{x} } }$

$={e}^{ \lim _{x \to \infty }\frac{ \frac{1}{1 + \frac{1}{x} } \frac{ - 1}{ {x}^{2} } }{ - \frac{1}{ {x}^{2} } } }$

$= {e}^{ \lim_{x \to \infty } \frac{x}{1 + x} }$

$= e$