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I am learning continuity. For proving the continuity of trigonometric functions why do we make such a replacement for x?I mean what is the intuition in making such a replacement? Is it like as we get h-->0 the limit f(x)-->sinx and thereby verifying continuity or is it more?enter image description here

Dhyaneshwar
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  • Hold on, I believe the replacement from the fourth line to fifth line in the derivation actually uses that $\sin(x)$ is continuous. But maybe the book isn’t doing this rigorously and don’t use epsilon-delta or the like. – Divide1918 Dec 15 '23 at 14:50

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For any function $f$ (not just trigonometric), to prove continuity at some particular number $c$ requires reasoning about the values of the function near $c$. That means looking at $f(c+h)$ when $|h|$ is small. If you know something about $f$ that helps you algebraically with $f(c+h)$ that's a natural place to start. It's what the author of your text is doing.

Ethan Bolker
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  • "If you know something about f that helps you algebraically with f(c+h) that's a natural place to start. " I don't understand what do you mean – Dhyaneshwar Dec 14 '23 at 17:37
  • If $x\to c$, then this means that $x-c=h$ is going to $0$, so we write $x=c+h$ and let $h\to 0$. The point is that for trig functions, exponential functions, and so on, we have an explicit formula for $f(c+h)$ (so-called addition formulas for the trig functions), so we can expand $f(c+h)$ in terms of that formula and see what happens as $h\to 0$. Same thing for $f(x)=x^2$. You have $(c+h)^2 = c^2 + 2hc + h^2$, for example. – Ted Shifrin Dec 14 '23 at 18:14
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Think about it this way. For a function f(x) to exist at x = a, the Left Hand limit and Right Hand limit must be equal to each other (and also equal to the value of the function defined at that point). Which we can write as:

$$\lim_{x \to a^{+}} f(x) = \lim_{x \to a^{-}} = f(x) $$

Now what is $a^{+}$ and $a^{-}$? Is it a valid number where you can do some elementary operations? No. It is simply a notation to say from which direction you approach the point. To fix that I'll replace $a^{+}$ with $a+h$, and $a^{-}$ with $a-h$ and I'll make sure that h approaches zero.

But in your example, they have only given the result of the Right Hand Limit, where in this case it is sufficient. For some arbitrary function g(x), to check the continuity at some point 'a', it is recommended to find both the LHL and RHL.

Akilan SS
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Assuming you're not asking "is this valid?", but "why do we bother?", it's because it makes the math simpler! Clearly

$$\sin(x) = \sin( c + (x-c)),$$

and, remembering that any $\lim_{x \rightarrow c}$ involves making $|x-c|$ small, we just decide to use a new variable, "$h$", to represent the quantity $(x-c)$, and examine what happens when $|h|$ gets small.

You could take all the computations done and back-replace $h$ with $(x-c)$, and the argument would be just as valid, but messier and harder to track.

I'll add that this is not limited to the types of functions you've listed, or to computing limits. The following two set-ups are equivalent, if we think of $r$ designating some point:

$$\text{Let's consider }s \text{ near } r...$$ and $$\text{Let's consider }r+t \text{ for small } t...$$

and people will use whichever is more convenient.

JonathanZ
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