Limits

Calculus Level 2

Find $[\displaystyle \lim _{x\to 0}100 \frac{\sin x}{x}]$ where [.] denotes Greatest Integer Fuction.

1 99 0 100

1 solution

Вук Радовић
Jan 30, 2015

Using Taylors series formula you get for small $x$ that $\sin x =x$ then result is $100$

Actually, even for small $x$ , $\sin x \lt x$ which can be verified by checking the Taylor series of $\sin x$ and its behavior about $x=0$ . Due to this, an important thing to be noted here is that the answer would've been $99$ , had the floor function been inside the limit.

Prasun Biswas - 6 years, 4 months ago

Yes you are absolutely right. The answer is 99 and not 100.

Anurag Pandey - 6 years, 1 month ago

Actually, no! The answer is $100$ here because the limit is inside the floor. Here's a depiction:

$\left\lfloor\lim_{x\to 0}100\cdot\frac{\sin x}{x}\right\rfloor=\lfloor100\rfloor=100$

$\lim_{x\to 0}\left\lfloor100\cdot\frac{\sin x}{x}\right\rfloor=\lim_{x\to 0} 99=99$

So, we can conclude that,

$\left\lfloor\lim_{x\to 0}100\cdot\frac{\sin x}{x}\right\rfloor\neq \lim_{x\to 0}\left\lfloor100\cdot\frac{\sin x}{x}\right\rfloor$

Here, $\lfloor\cdot\rfloor$ depicts the greatest integer function (a.k.a., floor function).

Prasun Biswas - 6 years, 1 month ago

@Prasun Biswas – Okay...so the answer changes in this two cases..thanks.

Anurag Pandey - 6 years, 1 month ago

Its answer is 99.You are wrong.

Himanshu Goel - 6 years, 3 months ago

Yes, the answer would have been 99 Since for small x sinx<x

Ansh Sapra - 6 years, 2 months ago

Answer will be 100 only. Floor of( 100 lim sinx/x) will be 100 But 100 lim floor(sinx/x) will be 99.

Aashay Agarwal - 4 days, 4 hours ago

Limits

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