اهلا

For a limit to even exist and answer this question, we must check the limit as x approaches zero from the left (values greater than zero) and from the right (values less than zero). This is the definition of the limit in this context.
From the left, we can plug in any number greater but very close to zero to the function, such as 0.000000001. 1/0.000000001 will be a very huge positive number. The end behavior of the function will still yield a positive large number if we plug in even closer values of zero from the left, or positive infinity.
From the right, we can plug in any number less than but very close to zero to the function, such as -0.000000001. 1/-0.000000001 will be a very huge negative number. The end behavior of the function will still yield a negative large number if we plug in even closer values of zero from the right, or negative infinity.
As a result, the left and right limits do not equal. Therefore, the final answer is that limit of 1/x as x approaches zero is DNE (do not exist) and/or that the function diverges.
The limit of [1/x] as x approaches 0 doesn’t exist.
The first reason for this is because left and right hand limits are not equal. Because 0 cannot be in the denominator there is a vertical asymptote at x=0.
This is an odd function meaning that it is symmetrical over the origin.
The limit of [1/x] as x approaches 0 from the right is equal to infinity. Since this function is symmetrical over the origin the limit of [1/x] as x approaches 0 from the left is equal to negative infinity. Since the right and left hand limits are not equal the limit doesn’t exist (this is the 1st reason)
Even if the limit on both sides did equal to infinity or negative infinity, infinity is not a real number. Since infinity isn’t a real number the limit cannot exist.
In no case does the limit technically exist, because limits are numbers. Limits don’t give you the right language to talk about this situation accurately, but:
is simply undefined.
These are shown in the video below.
Does Limit x ->0 (1/|x|) exist?
Taking the limit of 1/x as x goes to infinity, does the function approach 0, or does it reach 0?
Why does ?
The limit of 1/x as x approaches to zero doesn't exists mathematically as we say the
limit x->a[f(x)] exists if the Right Hand Limit and Left Hand Limit at x=a for f(x) are equal. The right hand limit is the limit when the limiting value is slightly greater than the tending constant and in left hand limit the value is slightly less than the constant.
Here, RHL
Ltx->0+(f(x))=Ltx->0+(1/x)=+oc
LHL
Ltx->0-(f(x))=Ltx->0-[1/x]=-oc
Here the limit value obtained from both the values are not equal as +oc≠-oc.
Hence the limit value we need to evaluate doesn't exists.
Ltx->0(1/x) does not exists.
We cannot evaluate this limit as it doesn't exists.
There are several possible answers. Some say infinity is not a number and therefore the limit doesn’t exist. Some treat infinity like a number, but say the limit doesn’t exist because you get + or - infinity depending whether x is positive or negative. Sometimes (especially in complex analysis) we like to identify all types of infinites and say the limit is infinite.
For the first type of person, when they say a limit is infinite, that is just a shorthand for “the limit doesn’t exist but the values become arbitrarily large as x tends to zero”.
Edit: I have since realised that the square brackets probably indicate the integer part function. I don’t think that affects the answer, but the function [1/x] is no longer continuous.
Plot the graph of
Look at the behavior of the function around
You will see that they move in opposite directions. This states that the limit does not exist.
The limit of [1/x] as x approaches 0, is infinity.
Try the following on your Calculator:
1/0.1 =10; 1/0.01 =100; 1/0.001 =1000;
1/0.000001 =1000000; You can see where you are heading to. Bigger and bigger quotient.
You will learn in school that approaching 0 from the left leads to -inf. And from the right to +Infinity. It is represented in a Cartesian plot:
Zero is the center point of the Cartesian coordinates system.
This link can help you Press here