# Find $\lim_{x \to 0}\frac{\cos 2x-1}{\cos x-1}$ without L'Hopital's rule.

$$\lim_{x \to 0}\frac{\cos 2x-1}{\cos x-1}$$ I have found the above limit using L'Hopital's rule but since this rule is not given in the book so I'm supposed to do it without using this rule.

I know $$\lim_{x \to 0}\frac{1-\cos x}{x}=0$$

I tried to get something of the form of the above limit but I failed to do so.

Kindly help me solve this problem without using L'Hopital's rule.

• Have you tried using trigonometric identities to make it less daunting? I believe there is one that will help. – Terra Hyde Jul 15 '15 at 3:17
• @TerraHyde Sir are you saying to use $cos(2x)=2cos^2(x)-1$. – Singh Jul 15 '15 at 3:19
• In particular, you can use a trig identity to get rid of the 2 inside the cosine in the numerator! – Trevor J Richards Jul 15 '15 at 3:19
• @Singh Yes, I was suggesting exactly that. Then it becomes very readily solvable. – Terra Hyde Jul 15 '15 at 3:23

Recall $\cos(2x)=2\cos^2(x)-1$ so we may rewrite as

$$\lim\limits_{x\to 0} 2\frac{\cos^2(x)-1}{\cos(x)-1}=\lim\limits_{x\to 0} 2\frac{(\cos(x)-1)(\cos(x)+1)}{\cos(x)-1}=2(\cos(0)+1)=4$$

Since others have already presented a great way forward, I thought that it would be instructive to present an alternative approach. To that end, we use

$$\cos x=1-\frac12 x^2+O(x^4)$$

to write

\begin{align}\frac{\cos 2x-1}{\cos x-1}&=\frac{\left(1-\frac12 (2x)^2+O(x^4)\right)-1}{\left(1-\frac12 x^2+O(x^4)\right)-1}\\\\&=\frac{-2x^2+O(x^4)}{-\frac12 x^2+O(x^4)}\\\\&=4+O(x^4)\to 4\end{align}

• this is equivalent to using l'Hôpital... – hjhjhj57 Jul 15 '15 at 5:14
• @hjhjhj57 Well, it isn't really equivalent. We don't have the appearance of $\frac{-2\sin 2x}{-\sin x}$ herein, which by the way would require a second round of LR to get to $\frac{-4\cos 2x}{-\cos x}$. Asymptotic expansions are not equivalent. – Mark Viola Jul 15 '15 at 5:28
• It's equivalent in the sense that both methods require the use of derivatives, in fact, you can see l'Hôpital as a simple corollary of this approach. – hjhjhj57 Jul 15 '15 at 5:49
• Not quite. First, if the cosine function is DEFINED in terms of its Taylor series, then we never formally took a derivative. And there are other ways to obtain the bounds used herein. On the second point ... not quite... the proof of LR typically relies on the extended MVT for which $\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(\xi)}{g'(\xi)}$ under the usual hypothesis. ;-)) – Mark Viola Jul 15 '15 at 6:08

Hint: try to expand cos 2x first.

Remember cos 2x = cos² x - sin² x, or cos 2x = 2cos² x - 1. You can continue solve this problem by using this identity.

$\dfrac{1-\cos (2x)}{1-\cos x} = \dfrac{2\sin^2x}{2\sin^2\left(\frac{x}{2}\right)}=4\cdot \left(\dfrac{\sin x}{x}\right)^2\cdot \left(\dfrac{\dfrac{x}{2}}{\sin\left(\dfrac{x}{2}\right)}\right)^2\to 4\cdot 1\cdot 1=4$, as $x \to 0$.