I've got this limit: $$\displaystyle\lim_{x\to 0} \frac{\ln(1-x)-\sin x}{1-\cos^2 x}$$ and the problem is that it doesn't exist. But I am not very perceptive and I didn't avoid catching in a trap and I started to trying solve this with L'Hôpital's rule. And my question is: are there any ways to notice that given limit doesn't exist in time? If I had been given such a limit on a test, what is the ideal way to solve it?
Evaluate $\lim_{x\to 0} (\ln(1-x)-\sin x)/(1-\cos^2 x)$
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6Mr. Taylor has a way. – 2012-07-07
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10Are you watching the movie *Mean Girls*? See a discussion [there](http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf450). – 2012-07-07
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0@did, actually I've seen this movie yesterday and this limit interested me :-) thank you very much for the link! – 2012-07-07
4 Answers
I am seeing the limit approaching -∞ if x approaches 0 from the positive side, and +∞ if x approaches 0 from the negative side.
I used L'Hôpital's rule one time. The limit is then of the form -2/0. The sign of the limit changes depending if x approaches from the positive side or negative side.
A possible way is to rewrite it as
$$\lim_{x\rightarrow 0}\frac{\ln (1-x)-\sin x}{1-\cos ^{2}x} =\lim_{x\rightarrow 0}\frac{\dfrac{\ln (1-x)}{\sin x}-1}{\sin x}\tag{0}$$
and evaluate by L'Hôpital's rule
$$\lim_{x\rightarrow 0}\frac{\ln (1-x)}{\sin x}=\lim_{x\rightarrow 0}\frac{\frac{-1}{ 1-x }}{\cos x}=-1.\tag{1}$$
Consequently,
$$\lim_{x\rightarrow 0}\frac{\ln (1-x)-\sin x}{1-\cos ^{2}x}=\infty .\tag{2}$$
Note: In view of Marvis' comment I add that this limit is $\infty$ without sign. The side limits depend on the denominator of $(0)$, since its numerator is $-2$ in the limit. For $x>0$ $\sin x>0$ and for $x<0$ $\sin x<0$. Hence
$$\lim_{x\rightarrow 0^+}\frac{\ln (1-x)-\sin x}{1-\cos ^{2}x}=-\infty. \tag{2a}$$
$$\lim_{x\rightarrow 0^-}\frac{\ln (1-x)-\sin x}{1-\cos ^{2}x}=+\infty. \tag{2b}$$
Added: Another way to compute $(1)$ is to write
$$\frac{\ln (1-x)}{\sin x}=\frac{\ln (1-x)}{x}\cdot \frac{x}{\sin x}$$
and use the elementary limits
$$\lim_{x\rightarrow 0}\frac{\ln (1-x)}{x}=-1,$$
$$\lim_{x\rightarrow 0}\frac{x}{\sin x}=1.$$
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0-1: since $\lim_{x \to 0^+}$ is $-\infty$ and $\lim_{x \to 0^-}$ is $\infty$ – 2012-07-07
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0@Marvis: I wrote $\infty$ without sign just because of that. – 2012-07-07
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0@Marvis In my notation above $\infty$ does not mean $+\infty$. I added a note with the two side limits. – 2012-07-07
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0Oh Ok. I will remove my down-vote. – 2012-07-07
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0@Marvis Thanks! – 2012-07-07
\begin{align} f(x) & = \dfrac{\log(1-x) - \sin(x)}{\sin^2(x)} = \dfrac{\left(-x - \dfrac{x^2}2 - \dfrac{x^3}3 - \cdots \right) - \left( x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots \right)}{\left( x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots \right)^2}\\ & = \dfrac{-2x + \mathcal{O}(x^2)}{x^2 + \mathcal{O}(x^3)} = -\dfrac{2+\mathcal{O}(x)}{x+\mathcal{O}(x^2)} \end{align} Hence, $$\lim_{x \to 0^+} f(x) = - \infty$$ $$\lim_{x \to 0^-} f(x) = \infty$$
Maclaurin series approach.
$\ln(1-x)\sim -x$, $1-\cos^2 x\sim x^2/2$ and $\sin x\sim x$ for $x \to 0$. Then:
$$\frac{\ln(1-x)-\sin x}{1-\cos^2 x}\sim \frac{-2x}{x^2/2}\to\infty$$
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0Can $\sim$ statements be added/subtracted? – 2012-07-07
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0Sorry, I don't understand. What do you mean? – 2012-07-07
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0In this case, I think yes, but in general it might be better to justify by using o/O notation? – 2012-07-07
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0It's true that $a\sim b$ and $c\sim d$ imply $a/c\sim b/d$. However, you seem to have combined the $\sim$ statements for $\ln(1-x)$ and $\sin x$ by subtracting the latter from the former. I feel this doesn't work in general, and big-$O$ notation would be justified, as OJ says. – 2012-07-07
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0If I had $\ln(1+x)-\sin x$ I could not use $\ln(1-x)\sim -x$ etc. – 2012-07-07
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0What you need to do is to establish the leading terms of the series expansion of numerator and denominator, because in the limit the other terms are irrelevant. – 2012-07-07
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0As $x\to0$, we have $1+1/x\sim 2+1/x$. We also have $1/x\sim1/x$ However, subtracting the latter from the former gives $1\sim2$. In general you cannot subtract one $\sim$ statement from another and get a true claim. This is my point. – 2012-07-07
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5Not that it matters, but $1-\cos^2(x)\sim x^2$ – 2012-07-07