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$$\lim_{x\rightarrow0}\frac{\ln(\cos x)}{x^2}$$

I've tried this

$$\lim_{x\rightarrow0}\left(\frac{\ln(\cos x)+1}{\cos x}\cdot\frac{\cos x}{x^{2}}-\frac{1}{x^{2}}\right)$$

but $\dfrac{1}{x^2}$ is goint to infinity. Please answer, how to solve it using easy methods. Thanks in advance!

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By easy methods do you mean without using l’Hospital’s rule or series? – Brian M. Scott Jan 11 '13 at 18:15
Without l'Hospital's rule (we don't know it yet), we know series. – Steve Jan 11 '13 at 18:16
@Steve: If, by chance, you have some knowledge about Taylor Series, you should be able to grasp l'Hospital's Rule easily; both contain the concept of the derivative. – Parth Kohli Jan 11 '13 at 18:18
@DumbCow I'm preparing for exam, and we aren't allowed to use L'Hospital on it. We will be learning about it in the next month. Thanks for advice! – Steve Jan 11 '13 at 18:19
@BrianM.Scott I was thinking about series expansion, $cos x = 1 - x^2/2 + x^4/4! - x^6/6! +...$, but don't see how it help. – Steve Jan 11 '13 at 18:22
up vote 6 down vote accepted

First write $\ln(\cos x)$ as $\frac{1}{2}\ln(1-\sin^2 x)$, then see the when $x\to 0$, $\sin^2 x$ is very small. We know that when the function $\alpha(x)$ is very small, then $\ln(1+\alpha(x))\sim\alpha(x)$ so $$\ln\left(\cos x\right)=\frac{1}{2}\ln(1+(-\sin^2 x))\sim \frac{-1}{2}\sin^2(x)$$

Now take your limit with this fact again.

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So to make a formal prof, I need to estimate this value from left and right side, and show that both are converging to -1/2? That mean, I need to use x/x+1 <= ln(1+x) <= x for x>-1? – Steve Jan 11 '13 at 18:34
@Steve: Besides to caloric's answer note that for all $x>0$ we have $$x-\frac{1}{2}x^2<\ln(x+1)<x$$ – Babak S. Jan 11 '13 at 18:41
+1 (re: edit - just a stray, extra parentheses removed!) – amWhy Jan 11 '13 at 19:04
@amWhy: You made me pleased, amWhy. Thanks – Babak S. Jan 11 '13 at 19:09

Hint#1: You can use these series (you can find them here) $$\cos(x)\approx1-\frac{x^2}{2}+o(x^2)$$ $$\ln(1-x)\approx -x+o(x)$$

Here I used O notation

After the consistent application of these series, you will get the correct answer $-\frac{1}{ 2}$


Hint#2 $$\ln(\cos(x))\approx\ln\left(1-\frac{x^2}{2}+o(x^2)\right)\approx-\frac{x^2}{2}+o\left(-\frac{x^2}{2}+o(x^2)\right)\approx-\frac{x^2}{2}+o(x^2)$$

All properties of $o(f)$ you can find here (in section "Little-o notation")

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I've tried but don't see it. Please give me more hints. – Steve Jan 11 '13 at 18:27
I edited my post, check it. – Oiale Jan 11 '13 at 18:37
Now I understand, thanks! – Steve Jan 11 '13 at 18:39
You are welcome – Oiale Jan 11 '13 at 18:42

Solution with no De-L'Hopital, Taylor Expansions or assymptotic equalities but assuming the limit exists:

First prove that $$\lim_{x\rightarrow0}\frac{\ln(\cos x)}{x}=0$$ This follows from the fact that $$\lim_{x\rightarrow0}\frac{\ln(\cos x)}{x}=\lim_{x\rightarrow0}\frac{\ln(\cos x)-\ln(\cos 0)}{x}=(\ln(\cos x))^{\prime}(0)$$ Now define $$f(x)=\begin{cases}\frac{\ln(\cos x)}{x}& x\neq 0\\ 0& x=0\end{cases}$$ $f$ is continuous everywhere and differentiable at least in $\mathbb{R}^*$ Observe that $$\lim_{x\rightarrow0}\frac{\ln(\cos x)}{x^2}=\lim_{x\to 0}\frac{f(x)-f(0)}{x}$$ and so we need to evaluate $f^{\prime}(0)$ (after proving its existence). We state: $$\lim_{x\to 0}\frac{f(x)-f(0)}{x}=\lim_{x\to 0}f^{\prime}(x)$$ Indeed, by the Mean Value Theorem for $x>0$, $\exists \xi_x\in (0,x)$ so that $$f^{\prime}(\xi_x)=\frac{f(x)-f(0)}{x}$$ Letting $x\to 0^+$, $\xi\to 0^+$ and so $$\lim_{x\to 0^+}f^{\prime}(x)=\lim_{x\to 0^+}\frac{f(x)-f(0)}{x}$$ Similarly for $x<0$ (the above is done assuming $\lim_{x\to 0}f^{\prime}(x)$ exists).

It remains to show $\lim_{x\to 0}f^{\prime}(x)$ exists and to evaluate it. Showing existence will not be trivial as the limit $\lim_{x\to 0}\frac{\cos x}{x^2}$ re-appears, albeit with a minus sign. Assuming the existence of $\lim_{x\to 0}\frac{\cos x}{x^2}$ however, one can easily evaluate it as $$\lim_{x\to 0}\frac{\cos x}{x^2}=\lim_{x\to 0}f^{\prime}(x)$$ (the last limit will have a term $-\lim_{x\to 0}\frac{\cos x}{x^2}$ and another easy limit. Pair up $\lim_{x\to 0}\frac{\cos x}{x^2}$ and you will be done

Of course if the existence is not assumed, one would have to use other trickery.

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That's very interesting answer, thank You! – Steve Jan 11 '13 at 18:47
@Steve Note that it is weaker than the others as it assumes existence of the limit. On the plus side the theorems involved should be familiar to you. – Nameless Jan 11 '13 at 18:48
You are a good tutor, Nameless. I am a bit jealousy of you. :) – Babak S. Jan 11 '13 at 18:59

(using L'Hopital rule) $$\lim_{x\rightarrow0}\frac{\ln(\cos x)}{x^2}=\lim_{x\to 0}\frac{-\tan x}{2x}=\lim_{x\to 0}\frac{-1}{2\cos^2x}=-\frac{1}{2}$$

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The OP made it clear that s/he cannot use L'Hospital. Of course, it can't hurt to demonstrate (foreshadow) what is to come :-) – amWhy Jan 11 '13 at 18:37

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