# Evaluating a trigonometric limit

What is the limit as $x$ approaches $0$ of: $$\frac{\sqrt{1+\tan x} - \sqrt{1+\sin x}}{1+x-\cos x}?$$

We cannot use L'Hôpital's rule or anything advanced like Taylor series. I reduced it to, by considering the numerator's conjugate:

$$\frac{1}{2}\lim_{x \to 0}\frac{\tan x - \sin x}{1+x-\cos x}$$

EDIT: Thinking carefully, I think I can simplify further:

$$\frac{1}{2}\lim_{x \to 0}\frac{\frac{\tan x - \sin x}{x}}{\frac{1-\cos x}{x}+1}=\frac{1}{2}\lim_{x \to 0}\frac{\tan x - \sin x}{x}$$

And perhaps the solution:

$$\frac{1}{2}\lim_{x \to 0}\frac{\tan x - \sin x}{x}=\frac{1}{2}(\lim_{x \to 0}\frac{\tan x}{x} - 1)=\frac{1}{2}(\lim_{x \to 0}\frac{\sin x}{\cos x \cdot x}) - 1=\frac{1}{2}(1- 1)=0$$

Is this correct?

The title of the question is a bit intimidating as this limit involves very basic trigonometric manipulation and standard limits like $\lim\limits_{x \to 0}\dfrac{\sin x}{x} = 1$. Here goes the simple solution \begin{aligned}L &= \lim_{x \to 0}\frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{1 + x - \cos x}\\ &= \lim_{x \to 0}\frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{1 + x - \cos x}\cdot \frac{\sqrt{1 + \tan x} + \sqrt{1 + \sin x}}{\sqrt{1 + \tan x} + \sqrt{1 + \sin x}}\\ &= \lim_{x \to 0}\frac{\tan x - \sin x}{(1 + x - \cos x)(\sqrt{1 + \tan x} + \sqrt{1 + \sin x})}\\ &= \frac{1}{2}\lim_{x \to 0}\frac{\tan x - \sin x}{1 + x - \cos x}\\ &= \frac{1}{2}\lim_{x \to 0}\tan x\cdot \frac{1 - \cos x}{1 + x - \cos x}\\ &= \frac{1}{2}\lim_{x \to 0}\tan x\cdot \frac{2\sin^{2}(x/2)}{x + 2\sin^{2}(x/2)}\\ &= \lim_{x \to 0}\tan x\cdot \dfrac{\dfrac{\sin^{2}(x/2)}{(x/2)^{2}}\cdot(x/2)^{2}}{x + 2\dfrac{\sin^{2}(x/2)}{(x/2)^{2}}\cdot(x/2)^{2}}\\ &= \lim_{x \to 0}\tan x\cdot \frac{x}{4}\cdot\dfrac{\dfrac{\sin^{2}(x/2)}{(x/2)^{2}}}{1 + \dfrac{x}{2}\cdot\dfrac{\sin^{2}(x/2)}{(x/2)^{2}}}\\ &= 0\cdot 0\cdot\dfrac{1}{1 + 0\cdot 1} = 0\end{aligned}

$$\frac{\sqrt{1+\tan x} - \sqrt{1+\sin x}}{1+x-\cos x}\sim_0\frac{\sqrt{1+x+\frac{x^3}3} - \sqrt{1+x-\frac{x^3}6}}{x+\frac{x^2}2}\sim_0\frac{\frac{1}{2}\left(\frac{x^3}3+\frac{x^3}6\right)}{x}\sim_0\frac14 x^2$$ so the desired limit is $0$.

Edit Without Taylor series the calculus is tedious: let $$f(x)=\sqrt{1+\tan x} - \sqrt{1+\sin x}$$ and $$g(x)=1+x-\cos x$$ then $$\lim_{x\to0}\frac{\sqrt{1+\tan x} - \sqrt{1+\sin x}}{1+x-\cos x}=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}\frac{x-0}{g(x)-g(0)}=\frac{f'(0)}{g'(0)}$$ and with painful calculus we find that $f'(0)=0$ and easily $g'(0)=1$ so we conclude.

• You can't use taylor series as OP said. – Euler....IS_ALIVE Feb 19 '14 at 6:17
• What I have done in the edit is correct? And I like your second method :) – Sawarnik Feb 19 '14 at 6:33
• @Sawarnik Yes correct! – user63181 Feb 19 '14 at 6:35
• @SamiBenRomdhane Thanks :) – Sawarnik Feb 19 '14 at 6:36
• You're welcome:) @Sawarnik – user63181 Feb 19 '14 at 6:37

$$\lim_{x\to 0}\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{1+x-\cos x}$$ Here the form is $(\frac{0}{0})$, so we can apply L'hospital rule. $$\lim_{x\to 0}\frac{\frac{1}{2\sqrt{1+\tan x}}\sec^{2}x-\frac{1}{2\sqrt{1+\sin x}}\cos x}{1+\sin x}=0.$$

• What about "cannot use LH" part? :) – Kaster Feb 19 '14 at 6:26
• Taylor Series and Lhopital's rule can't be used. – Euler....IS_ALIVE Feb 19 '14 at 6:28