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This question asks to find the $\lim_{x\to0}(x\tan x)^x$ . Ron Gordon and Maisam Hedyelloo make the substitution $ x\sim \tan x$ , and it works and they get the correct answer. However, if you try to make the substitution $x \sim \arcsin x$ into $\lim_{x \to 0} \large \frac {\arcsin(x)-x}{x^3}$ you get the wrong result of $0$, when in reality the limit is equal to $\frac 16$ . So when can you use this kind of substitution? Thanks.

P.S. I have asked a similar question here , and I thought I had the answer to the question but now I see that it is not complete.

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Ovi
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  • $$\lim_{x \to 0} \large \frac {\arcsin(x)-x}{x^3} = \lim_{x \to 0} \large \frac {x-\sin(x)}{(\sin (x))^3}= \lim_{x \to 0} \large \frac {x-\sin(x)}{x^3}\frac{x^3}{(\sin (x))^3}=\frac{1}{6}\cdot 1$$ The sin sub works in the second limit. – N. S. Jul 18 '13 at 16:28
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    You can do it when it will make no difference! And you find out when it makes no difference, what is safe to give away, by experience. After a while, one can look at many expressions and see that it would be mechanical to prove that the substitution makes no difference. At this stage, you should try to develop intuition, but also check in detail each time, unless you have done almost the same thing before. – André Nicolas Jul 18 '13 at 16:33
  • @AndréNicolas Ok thank you! – Ovi Jul 18 '13 at 16:53

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You can substitute whenever the function you are limiting is continuous at the limiting point.

The example with $\arcsin(x)$ that you gave isn't continuous at $x=0$ (it's not defined there).

By the way, you don't even get $0$ when you "substitute" into that equation: you get $\frac{0}{0}$ which is indeterminate.

rschwieb
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  • Why isn't $\arcsin x$ continuous at 0, and isn't $\lim_{x\to0} \frac{0}{x^3}=0?$ – user84413 Jul 18 '13 at 23:20
  • @user84413 You aren't allowed to pick which $x$'s you substitute: you have to do them all at once. – rschwieb Jul 18 '13 at 23:26
  • @user84413 and $\arcsin(x)$ is continuous at $0$. But that's not the entire function: $\arcsin(x)/x^3$ is the function to pay attention to as a whole. – rschwieb Jul 19 '13 at 01:34
  • I'm sorry; now I understand what you're saying about the arcsine function. I don't think the OP was asking about substituting the value of 0 in the limit, though; they wanted to know when substituting one function for another in the limit was valid, and that's what Andre's comment was addressing. – user84413 Jul 19 '13 at 17:56