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Určete

$$\lim\limits_{x \to 0}{\sqrt{1+\tan x}-\sqrt{1-\tan x}\over \sin x}$$

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Platí

$${\sqrt{1+\tan x}-\sqrt{1-\tan x}\over \sin x}=
{{1+ \tan x - 1 + \tan x}\over \sin x [\sqrt(1+\tan x) + \sqrt(1-\tan x)]]}= 
{2 \over \cos x[\sqrt(1+\tan x) + \sqrt(1-\tan x)]}$$

Podle věty o limitě spojité funkce se hledaná limita rovná 1.

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