What is $\lim_{x\to\infty} \sin x$?

What is

$$\lim_{x\to\infty} \sin x$$

?

I always thought it was undefined; however, Wolfram|Alpha says that

$$\lim_{x\to\infty} \sin x = -1 \text{ to } 1$$

Now, what is the correct answer? What does it even mean if the limit is not a single, distinct value, but rather an interval?

Other than that, am I right in assuming that

$$\liminf_{x\to\infty} \sin x = -1$$

and

$$\limsup_{x\to\infty} \sin x = 1$$ ?

• They mean that the values of $\sin(x)$ for $x\to+\infty$ accumulate to every value in $[-1,1]$. In particular, that implies the $\limsup$, $\liminf$ equations that you wrote, and that in the context of an introduction to calculus, the limit doesn't exist. – user561348 May 24 '18 at 12:37
• @arugula so is it in fact undefined? – user500664 May 24 '18 at 12:37
• Who knows what Alpha means, but I guess you could interpret it as "for every point $p$ between -1 and 1, there's a sequence tending to infinity, such that $\sin$ of that sequence is constantly $p$". And yes you're right about the sup and inf limits. – Steve D May 24 '18 at 12:38
• To find what Wolfram Alpha means may be difficult? But presumably this is the same answer as Mathematica itself, which does have extensive documentation you can look at. – GEdgar May 24 '18 at 12:45

Take note that the sine function takes values between $-1$ and $1$. When you are talking about a limit to $\infty$, it's an undetermined number, which is infinitely large. Now, taking into account the periodicity of the sine function, there is no possible way to determine a specific value, as it entirely depends on the nature of the "infinite" number.

More specifically, $-1 \leq \sin(x) \leq 1, \; \; \forall x \in \mathbb R$. This means that for any given $x$ over the real numbers, the sine function is bounded. Thus, all you can say about an undetermined infinite limit (it does not exist talking strictly mathematics), is :

$$-1 \leq \lim_{x \to \infty} \sin(x) \leq 1$$

What you mentioned though is indeed true :

$$\liminf_{x\to\infty} \sin x = -1$$

$$\limsup_{x\to\infty} \sin x = 1$$

• So it can be understood as "the limit doesn't exist, but if it existed it would be between -1 and 1", right? – user500664 May 24 '18 at 12:42
• Essentially, yes. The limit does not exist, but it surely is bounded. – Rebellos May 24 '18 at 12:43
• alright, thank you. I will accept your answer as soon as it will let me. – user500664 May 24 '18 at 12:43

The limit doesn't exist and it can be proved formally by 2 subsequences with different limits, that is

• $x_n=2n\pi+\frac{\pi}2 \to \infty \implies \sin(x_n)=1$

• $x_n=2n\pi+\frac{3\pi}2 \to \infty \implies \sin(x_n)=-1$

Yes what is true is that $\liminf=-1$ and $\limsup=1$ indeed

$$-1\le\sin x \le 1$$

and we have found 2 subsequences which tends to those limits.

• Thank you for the proof as well. Your answer is very good, but I promised Rebellos to accept his answer earlier, so I am sorry. Thank you very much though. – user500664 May 24 '18 at 12:55
• @ThomasFlinkow You are welcome! The choice of the answer to be accepted is absolutely up to you! Bye – user May 24 '18 at 13:05