Thread: a question about limits

doesn't limits define when f(x) becomes undefined(changes it's behavior and becomes dis continuous??) as x approaches a real number a??
i could not understand the following example:

( x ^ 2 + x - 2 ) / ( x - 1 ) = ( ( x - 1 ) ( x + 2 ) / ( x - 1) ) = x + 2 ,provided x does not equal 1.

it follows that the graphs of the equations (the first one) and (the second one ) are the same except for x = 1.

1- lim f(x) = 3 as x approaches 1 ,as the equation is f(x) x + 2

2- lim g(x) = 3 as x approaches 1 , as the equation is g(x) = ( x ^ 2 + x - 2 ) / ( x - 1 )

3- lim h(x) = 3 as x approaches 1 ,as the functions is:

h(x) = { g(x) if x is not equal 1 ,and 2 if x = 1

could not understand the piece-wise defined function ,isn't g(x) supposed to become undefined as x = 1 then how the limit of the h(x) = 3 ?? and what does 2 mean in this piece-wise defined function (h(x)) ??
as for f(x) it is a continuous function ,correct???and as for g(x) it is a dis continuous function at x = 1 ,correct?

and here is another question ,it says prove that :
lim f(x) = 1 /x as x approaches 0 does not exist.
doesn't f(x) becomes undefined as x = 0 ,then how the limit of f(x) as x approaches 0 does not exist?

When f(x) is continuous at x = a, limit and function value at a are the same.
In your example, where f(x) has a "perforation", the limit represents the value which would make f(x) continuous there, it would "fill the hole".

Have you gotten a rigorous definition of a limit (epsilon-delta perhaps?) and if so, do you fully understand it?

As for your last question, if f(x) is defined on a neighborhood of x = a, then the limit of f(x) for x approaching a exists if and only if upper and lower limit (or "right" and "left") exist and are equal. Use this on f(x) = 1/x to show that the limit doesn't exist.

thanks for the reply.the following is my understanding for a limit:
a limit defines when a function changes it's state , lim f(x) exits if f(x) takes real number values as x --> a .to say the a limit for a function exits ,it's right and left limits must exist and equal each other.that what i could understand ,anyway i'm having another problem :

A mail-order company adds a shipping and handling fee of $4 for any order that wighs up to 10 lb with an additional 40 cents for each pound over 10 lb.
(a)find a piecewise-defined function S for the shipping and handling fee on an order of x pounds.
(b)if a is an integer greater than 10, find:
lim S(x) as x approaches a from the left
lim S(x) as x approaches a from the right

My answear was:

(a) S(x) = { 4 : if x <= 10 , 4 + 0.40 (x - 10) : if x > 10
(b) 4 , 4

The correct answear is:
(a) S(x) { 4 :if 0 < x <= 10 , 4 + 0.4 [[ x - 9 ]] :if x> [[ x ]] and x > 10 , 4 + 0.4( x - 10 ) : if x = [[x]] and x > 10
(b) 0.4a ; 0.4( a + 1 )

couldn't understand what does [[]] mean ,i mean what does if x > [[ x ]] (i can't understand the second and the third function in S(x) and what is [[x - 9]] for) ??

Looks to me to be the Greatest Integer Function (correct me if I'm wrong) although I thought it was more like .
Are you aware of it?

i don't know what is the greatest integer function ,does it mean rounding a decimal number to an integer number ? i tried to search Wikipedia but it didn't work.sorry as i'm not native english and it's been a while since i opened a math book. the sign in the book is [[]] ,maybe a mistake but still don't know what this piecewise_defined function means

Originally Posted by mHadad

i don't know what is the greatest integer function ,does it mean rounding a decimal number to an integer number ? i tried to search Wikipedia but it didn't work.sorry as i'm not native english and it's been a while since i opened a math book. the sign in the book is [[]] ,maybe a mistake but still don't know what this piecewise_defined function means

Yeah, it rounds off the number having a rational part to the greatest integer which is less than it. It's like this.

thanks shubh ,i got it

I was not following this from the very beggining thus I do not know what is going on. But it is not necessarily to write,

The greater integer function is indepotent.
Meaning,

i have a question in limits ,it says:

According to theory of relativity, the length of an object depends on it's velocity v ( look at the equation down the page ) Einstein also proved that the mass m of an object is related to v by the formula :

m = m0 / ( sqrt. 1 - v ^ 2 / c ^ 2 )
where m0 is the mass of the object at rest.

Investigate lim m as v approaches m from the left.

L = L0 ( sqrt 1 - v ^ 2 / c ^ 2 )
:Where L denotes Length of a moving object, L0 length of this object at rest, velocity of the object, c speed of light

don't know i have found that it is a division by zero as the velocity of this object is approaching the speed of light ,so it means that a mass would be more less as it is approaching the speed of light but it can't reach the speed of light or else it's mass state would be undetermined?

Originally Posted by mHadad

i have a question in limits ,it says:

According to theory of relativity, the length of an object depends on it's velocity v ( look at the equation down the page ) Einstein also proved that the mass m of an object is related to v by the formula :

m = m0 / ( sqrt. 1 - v ^ 2 / c ^ 2 )
where m0 is the mass of the object at rest.

Investigate lim m as v approaches m from the left.

L = L0 ( sqrt 1 - v ^ 2 / c ^ 2 )
:Where L denotes Length of a moving object, L0 length of this object at rest, velocity of the object, c speed of light

don't know i have found that it is a division by zero as the velocity of this object is approaching the speed of light ,so it means that a mass would be more less as it is approaching the speed of light but it can't reach the speed of light or else it's mass state would be undetermined?

Investigate lim m as v approaches m from the left.

This is supposed to be the limit of m as v approaches c from the left?



The meaning behind this is that as an object approaches the speed of light its mass increases without limit. aka. It takes an infinite amount of energy to accelerate an object to the speed of light, so it can't be done. (Typically this argument is given in terms of the object's momentum, not mass, but the mass argument does the job just as well.)

A similar argument shows that the object's length goes to 0 as v goes to c. This is also a ridiculous statement, so again we see that it can't happen.

NOTE: In both of these cases we are speaking of an observer watching the object move with a speed v according to his/her reference frame. The mass and length of the object in its reference frame are still m0 and L0 respectively.

-Dan

what is the best, easiest and free computer graphing calculator that will draw me my function .

I would definitely say the graphing program Hacker showed me

Originally Posted by mHadad

what is the best, easiest and free computer graphing calculator that will draw me my function .

The best and easiest are not necessarily the same thing. If you look at the
graphs attached to my posts you will see that I am not using the same
tool at ImPerfectHacker and Quick et al. This is because the tool
I use is what I use for other things, and it is very much more powerfully than
that the other.

I use it because it is on all my machines, I am fluent in it, and it does
the job of plotting adequately.

If I had no such tool I would use ImPerfectHacker recommendation.
But for me the best and easiest is Euler.

RonL