Question

Why is the limit of int(x) as x approaches zero from the right side, 0?

Answer 1

What int() function do you mean, precisely?

Answer 2

int(x) means integral of x

Answer 3

My book says \[LIM X->0 INT(X)\]

Answer 4

\[\lim_{x \rightarrow 0^+}\int\limits_{}^{}x dx \]

Answer 5

Unless the book as said int means something different

Answer 6

I'm in BC calc, and we do AB too. We didn't get to integrals yet

Answer 7

I looked it up and it suggested it was the greatest integer function???

Answer 8

I checked the answer and it is 0

Answer 9

That's what I was wondering, I was suspecting the 'nearest integer' function, in which case:
\[I | I \in Z, |I-x| = \min{|J-x|} \forall J \in Z\]

Answer 10

For \(0 \le x \lt 1\), the value of the greatest integer function is 0. So the limit is zero.

Answer 11

In which case, for x in the neighbourhood of (-0.5,0.5), INT(x) is always equal to 0, thus for limit from the right, x in (0, 0.5) is 0, thus the limit is zero

Answer 12

My mind went directly to integrals because int is also used to mean integrals
well in latex they are

Answer 13

But the limit would have also been the same then

Answer 14

Ooooohh. I did not think about that!! Thank you very much!!

Answer 15

http://www.icoachmath.com/math_Dictionary/Greatest_Integer_Function.html

Answer 16

@myininaya I was thinking along the same lines, but was suspicious of the lack of other formatting!

Answer 17

Thanks again guys!!

Answer 18

You are welcome.

Answer 19

Can you compute what the limit would be of the greatest integer function if x approaches zero from the left?

Answer 20

-1?

Answer 21

Correct! Good job!

Answer 22

Thanks. So the limit is just the highest value of the function as you approach a certain value?

Answer 23

That is not the definition of limit in general. But the definition of "greatest integer function" at x is the highest integer less than or equal to x.

Answer 24

So when we approach the greatest integer function from the left, x is in the interval:
\(-1 \le x \lt 0\) and in that interval, the function has a value of -1 for ALL points in that interval. Therefore, the limit is -1.

Answer 25

Why is it only restricted to that interval? Isn't the greatest integer function infinite?

Answer 26

I should have been more specific. Here we are talking about the limit as x approaches 0 from the left and so we are interested in the interval [-1, 0).
If you look at the graph of the greatest integer function you will notice it is a stepwise function and it has the value of -1 in the interval [-1, 0), 0 in the interval [0, 1), 1 in the interval [1, 2), etc.

Answer 27

Why aren't we concerned with (-infinity,0] and just -1 to 0?

Answer 28

The domain of the function is (-infinity, infinity) but here we are interested in the limit as x approaches 0 from the left (I asked you this question) and from the right (the original problem). So the intervals of interest to us are: [-1, 0) and [0, 1).