Limit of f(x) = ((1+tan x)^0.5 - (1+sin x)^0.5)/x^3 as x → 0

[lkh1986]

Find limit of f(x) when x approaches 0.

Given that f(x) is (1/x^3){(1+tanx)^0.5 - (1+sinx)^0.5}

The given answer is 0.25, but can somebody show me the solutions? I try the conjugate, and nothing works. Then I try to substitute t=(1+tanx)^0.5 and others, can't work, too.

 

[TD]

If you multiply numerator and denominator with the complement expression, you get after simplifying:

$$\frac{{\frac{{\tan x - \sin x}}{{x^3 }}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }}$$

Replacing sin(x) and tan(x) by the first terms of their Taylor series arround 0 so that the difference isn't 0, is x³/2. So you get:

$$\mathop {\lim }\limits_{x \to 0} \frac{{\frac{{\tan x - \sin x}}{{x^3 }}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }} = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{2}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }}$$

 

[0rthodontist]

Find limit of f(x) when x approaches 0.

Given that f(x) is (1/x^3){(1+tanx)^0.5 - (1+sinx)^0.5}

Wait a minute, what is your f(x)?
Is it
$$\frac {\sqrt{1+tanx} - \sqrt{1+sinx}}{x^3}$$?