Approaching Zero

Calculus Level 2

$\boxed{\begin{array}{c|r:r:r:r:r} x & 0.1 & 0.01& 0.001& 0.0001& 0.00001\\ \hline \lfloor x \rfloor & 0 & 0 & 0 & 0 & 0 \end{array}}$

By looking at the table above, is it true that as $x$ approaches 0, then $\lfloor x \rfloor$ approaches 0 as well?
That is, is $\displaystyle \lim_{x\to0} \lfloor x\rfloor = 0$ correct?

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

It is correct It is not correct

1 solution

Agnishom Chattopadhyay
Aug 30, 2016

Relevant wiki: Floor Function

While this table is true:

$\boxed{\begin{array}{c|r:r:r:r:r} x & 0.1 & 0.01& 0.001& 0.0001& 0.00001\\ \hline \lfloor x \rfloor & 0 & 0 & 0 & 0 & 0 \end{array}}$

This table is true as well:

$\boxed{\begin{array}{c|r:r:r:r:r} x & -0.1 & -0.01& -0.001& -0.0001& -0.00001\\ \hline \lfloor x \rfloor & -1 & -1 & -1 & -1 & -1 \end{array}}$

In essence, the function does not approach the same limit from both sides.

This is why the limit does not exist.

Don't agree - by looking at the table (as ordered) it IS correct ( even if it's not correct in R). And note: the domain of definition isn't stated.

J T - 2 years ago

Yeah, but the limit says as x approaches 0, that is from both sides. For the two sided limit to exist, it must be the case that both one sided limits are equal!

Sotiris Simos - 1 month ago

Approaching Zero

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