Evaluate $\lim\limits_{x \to \infty} \sin(\frac{1}{x})^x$

Question

This in an exercise in my Analysis book in a section on L'Hopital's rules. $$\lim\limits_{x \to \infty} \left[\sin\left(\frac{1}{x}\right)\right]^x$$ Now it's an indeterminate of the form $0^\infty$ however I don't know how to solve this. I have tried the following:

$$y=\lim\limits_{x \to \infty} \left[\sin\left(\frac{1}{x}\right)\right]^x$$ $$\ln y=\lim\limits_{x \to \infty}\ln \left[\sin\left(\frac{1}{x}\right)\right]^x$$ $$\ln{y}=\lim\limits_{x \to \infty} x\ln{\sin\frac{1}{x}}$$ Now this is an indeterminate limit of form $\infty\cdot\infty$ which approaches $\infty$. However I may not write now that therefore $y=e^\infty=\infty$.

How do I write this out correctly?

Answer 1

For $x>6/\pi,$ $0<\sin(1/x) <1/2.$ For such $x$ we have $0 < \sin(1/x)^x < (1/2)^x \to 0.$ Therefore the limit is $0$ by the squeeze theorem.

Answer 2

PRIMER:

In THIS ANSWER, I developed the pair inequalities that is introduced in elementary geometry

$$\bbox[5px,border:2px solid #C0A000]{\theta\cos(\theta) \le \sin(\theta) \le \theta} \tag 1$$

for $0\le \theta \le \pi/2$.

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Using $(1)$ with $\theta=1/x$, we have

$$\frac{\cos(1/x)}{x}\le \sin(1/x)\le \frac1x$$

for $\frac2\pi \le x$. Hence, we can write

$$\left(\cos(1/x)\right)^{x} \frac{1}{x^x}\le \left(\sin(1/x)\right)^x\le \frac1{x^x}$$

whence application of the squeeze theorem yields the coveted limit

$$\bbox[5px,border:2px solid #C0A000]{\lim_{x\to \infty} \left(\sin(1/x)\right)^x=0}$$

Answer 3

Substitute $x=1/t$ and compute the limit of the logarithm: $$ \lim_{x\to\infty}\log\left(\left((\sin\frac{1}{x}\right)^x\right) =\lim_{x\to\infty}x\log\left(\sin\frac{1}{x}\right) =\lim_{t\to0^+}\frac{\log\sin t}{t}=-\infty $$ Thus your limit is $\lim_{u\to-\infty}e^u=0$.

Answer 4

In fact you already know that $|\sin(u)|\le1$ for all $u$, and when $u\to0$ this sinus also gets smaller.

So no matter what's really inside the sinus, it can be a much more complicated expression than just $u=1/x$, till it can become sufficiently small then you have :

$|\sin(u)|\le C<1$ and it is well known that $C^x\to0$ when $x\to+\infty$ and so is the whole expression.

For instance if you ask me the same question for $\sin(\frac{53+x^2-\ln(x)}{\sqrt{\pi^x}})^x$ then my answer will be exactly the same.

And it doesn't even need to tend towards $0$, we just need the quantity inside the sinus to be small enough [ i.e. $|u|\le\alpha$ with $\alpha<\frac{\pi}{2}$ so that $C<1$]. For instance $sin(\frac{3x^2+5x+31}{2x^2-7})^x$ will do.

Of course you can always do finer majorations, but being lazy in mathematics is not always bad ;