lim-x-0-e-x-e-x-2-x-2-

Question Number 87153 by jagoll last updated on 03/Apr/20

\lim_{x \to 0} \frac{e^{x} + e^{- x} - 2}{x^{2}}

Commented by john santu last updated on 03/Apr/20

\lim_{x \to 0} \frac{e^{x} - e^{- x}}{2 x} = \lim_{x \to 0} \frac{e^{x} + e^{- x}}{2} = 1

Commented by mathmax by abdo last updated on 03/Apr/20

w e h a v e e^{x} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + o (x^{3}) (x \to 0)

e^{- x} = 1 - \frac{x}{1!} + \frac{x^{2}}{2!} + o (x^{3}) \Rightarrow e^{x} + e^{- x} - 2 = x^{2} + o (x^{3}) \Rightarrow

\frac{e^{x} + e^{- x} - 1}{x^{2}} = 1 + o (x) \Rightarrow {l i m}_{x \to 0} \frac{e^{x} + e^{- x} - 2}{x^{2}} = 1

Answered by $@ty@m123 last updated on 03/Apr/20

\lim_{x \to 0} \frac{e^{x} + \frac{1}{e^{x}} - 2}{x^{2}}

\lim_{x \to 0} \frac{e^{2 x} + 1 - 2 e^{x}}{x^{2} . e^{x}}

\lim_{x \to 0} \frac{{(e^{x} - 1)}^{2}}{x^{2} . e^{x}}

\lim_{x \to 0} {(\frac{e^{x} - 1}{x})}^{2} \times \frac{1}{\lim_{x \to 0} e^{x}}

= 1^{2} \times 1

= 1