Finding limit when the change of variable is not one-to-one

I want to rigorously prove that L=limx→0sin−1xx=1\displaystyle L= \lim_{x \to 0} \frac{\sin^{-1}x}{x} =1, without using l’hospital rule.

When we let siny=x\sin y = x, can I write L=limy→0sin−1(siny)siny=limy→0ysiny=1\displaystyle L= \lim_{y \to 0} \frac{\sin^{-1}(\sin y)}{\sin y}= \lim_{y \to 0} \frac{y}{\sin y}=1 ?

Or, do I need an argument like

L=limy→kπsin−1(siny)siny=limy→kπy−kπsiny=limy→kπy−kπsin(y−kπ)=limz→0zsinz=1\displaystyle L= \lim_{y\to k\pi} \frac{\sin^{-1}(\sin y)}{\sin y}= \lim_{y \to k\pi}\frac{y-k\pi}{\sin y}=\lim_{y \to k\pi}\frac{y-k\pi}{\sin (y-k\pi)}= \lim_{z \to 0} \frac{z}{\sin z}=1?

What are the assumptions in each step above? For example, “if yy is close enough to kπk\pi, then siny\sin y is one-to-one. Otherwise, the first equality above might not be true” etc.

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You first idea is correct. You can make it more formal, setting y=arcsinxy=\arcsin x. B.t.w., preferably use the mathematical name of the function, not sin−1\sin^{-1}, which is the name for hand-held calculator, given the limited space available on a key.
– Bernard
2 days ago

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@Bernard: Although the notation sin−1\sin^{-1} has its shortcomings, it is a pretty standard notation in Indian math texts and when one learns this notation for the first time, it always comes with a warning that (−1)(-1) is not an exponent here, but rather the notation for inverse function so that students don’t write sin−1x=1/sinx\sin^{-1}x = 1/\sin x. The notation arcsin\arcsin appears to be without any shortcomings and perhaps more popular internationally.
– Paramanand Singh
2 days ago

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@Paramanand Singh: What I criticize about the ^–1 notation, is mainly that it tends to let beginners think the sine function is bijective. Furthermore arcsin, etymologically, is more telling than a generic notation.
– Bernard
2 days ago

  

 

@Bernard. If you are French, I totally understand your point. These are aong the amazing things : when at was at university, I never heard about csc(x)\csc(x) or sec(x)\sec(x) forgetting all the other. Cheers.
– Claude Leibovici
2 days ago

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@imranfat: I learnt cotantgent, not the others. We knew these existed, of course, but they were considered old stuff, useful only for land-surveyors. Just as everyone knows the decimal logarithm, but no one sees the necessity to study it per se as a function
– Bernard
yesterday

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