use-defintion-to-prove-that-lim-x-0-1-x-1-x-x-1-

Question Number 70025 by Tony Lin last updated on 30/Sep/19

u s e ε - δ d e f i n t i o n t o p r o v e t h a t

\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} = 1

Commented by mind is power last updated on 01/Oct/19

= l i \underset{t \to 0}{m} \frac{\sqrt{1 + s i n (t)} - \sqrt{1 - s i n (t)}}{s i n (t)}

= \underset{x \to 0}{\lim_{}} \frac{1}{c o s (\frac{t}{2})}

\forall ε > 0 \exists η > 0

∣ x ∣⩽ η \Rightarrow∣ f (x) - 1 ∣⩽ ε

1 - ε ⩽ \frac{1}{c o s (\frac{t}{2})} ⩽ 1 + ε

\Rightarrow \frac{1}{1 - ε} ⩾ c o s (\frac{t}{2}) ⩾ \frac{1}{1 + ε}

\Rightarrow \frac{t}{2} ⩽ a r c o s (\frac{1}{1 - ε)}

\Rightarrow t ⩽ {2 \cos}^{- 1} (\frac{1}{1 - ε}) = η_{ε}