The priority of limits

Question

How are the two expressions different?

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor$$

and $$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor$$

If limit is inside the floor function, Do I have to apply the limits first?

If this is the case, then, $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor=0$$

$$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor=1$$

Am I solving this right?

Also how can I calculate,

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor$$

Thank you.

Answer 1

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor$$ In this first, you have to take the floor of the function then apply the limit on its floor. $$\frac{\sin{x}}{x}< 1$$ when $x \to 0$ $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor=0$$

$$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor$$

Here you have to first calculate the limit then take the floor of it. You know that this limit is 1 thus floor of 1 is 1. $$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor=1$$

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor$$

Series expansion at $x=0$ gives $$1+\frac{x^2}{6}+\frac{31x^4}{360}+O(x^6)$$

$$\frac{\sin{x}\cdot \tan{x}}{x^2}\ge 1$$

$$\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor=1$$ thus $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor=1$$

Answer 2

The first expression says:

• Take the function $f(x) = \lfloor\frac{\sin(x)}{x}\rfloor$.
• Take the limit $$\lim_{x\to 0} f(x)$$

The second expression says:

• Take the function $f(x) = \frac{\sin(x)}{x}$.
• Take the limit $$L=\lim_{x\to 0} f(x)$$.
• Calculate $\lfloor L\rfloor$

The two expressions do not need to be the same.

Answer 3

Yes you are right indeed since

$$\frac{\sin x}{x}<1 \implies \bigg \lfloor\frac{\sin{x}}{x}\bigg\rfloor=0 \implies \lim_{x\to0} \bigg \lfloor\frac{\sin{x}}{x}\bigg\rfloor=0$$

and since

$$\lim_{x\to0} \frac{\sin{x}}{x}=1 \implies \bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor=1 $$

For the latter for $x$ sufficiently small we have

$$1<\frac{\sin{x}\cdot \tan{x}}{x^2}<2$$

indeed $\sin x>x-\frac{x^3} 6$ and $\tan x>x+\frac{x^3} 3$ and

$$\frac{\sin{x}\cdot \tan{x}}{x^2}>1+\frac{x^2}6-\frac{x^6}{18}$$

therefore

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor=1$$

Answer 4

Here's fact 4

If $f(x)$ is continuous at $a$ and $\mathop {\lim }\limits_{x \to a} g\left( x \right) = b$ then:

$\mathop {\lim }\limits_{x \to a} f\left( {g\left( x \right)} \right) = f\left( {\mathop {\lim }\limits_{x \to a} g\left( x \right)} \right)$

Floor function isn't continuous, so you should apply floor function first, before calculating limit.

Let's calculate

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor$$

Okay, the first thing I want to do it's to solve: $$\lim_{x\to0}\frac{\sin{x}\cdot \tan{x}}{x^2} = \lim_{x\to0} \frac{\sin{x}}{x}{\frac{\sin{x}}{x}}{\frac{1}{\cos{x}}} = [\lim_{x\to0}\frac{\sin{x}}{x} = 1] = \lim_{x\to0}\frac{1}{\cos{x}} = 1$$

So, $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor = \lim_{x\to0}\bigg\lfloor\frac{1}{\cos{x}}\bigg\rfloor $$ $$\cos{x} \le 1 $$$$\frac{1}{\cos{x}} \ge 1$$

$$x < \arccos(2) => \cos{x} < \frac{1}{2} => \frac{1}{\cos{x}}< 2$$

It means that in the neighborhood of 0 with radius $\delta < arccos(2)$ function $\bigg\lfloor\frac{1}{\cos{x}}\bigg\rfloor \equiv 1$

And we can conclude that

$$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}\cdot \tan{x}}{x^2}\bigg\rfloor = \lim_{x\to0}\bigg\lfloor\frac{1}{\cos{x}}\bigg\rfloor = 1 $$