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about limit
- Thread starter Anikets
- Start date
Hi again,
Ok thanks for the clarification.
BTW i always did a "colon" plus "hyphen" plus "right paren" and i got a smily i think:
EDIT: yes it worked again here.
djsfantasi
- Joined Apr 11, 2010
- 9,237
Testing a Smiley to see if he's happy.
Just had to click at the desired position after clicking on the Smiley icon in the edit bar.
Just had to click at the desired position after clicking on the Smiley icon in the edit bar.
Now I presume you are talking about "Limit of a function at a point" . If that is correct then here is the truth, the whole truth and nothing but the truth, which in this case, is easy to find from many sources but let us break it down to a bit.
Let us for the sake of simplicity assume we are talking about functions that take a single real number (as opposed to complex number) as its argument and provides a single real number as its value.
Definition : A real number L is said to be the limit of a real valued function f(x) at a point A, if for every number epsilon > 0, there exits a number delta > 0 such | f(x) - L | < epsilon when ever | x - A | < delta .
What this means is that the limit "L" of a function if it exists, is defined point wise i.e. for each point where the function is defined. The limit may be different at different points. Further this "L" has a special property, roughly stated as follows
1) if you give my any positive number epsilon no matter how small
2) then I can find you a positive number delta such that
3) for all points within a distance delta of A
4) the value of the function is within a distance epsilon of L
5) the value of delta i will give you will depend on the value of epsilon you give me. I will change delta if you change epsilon.
This is a very precise albeit mind bending way to define the notion of the value a function "tends to" at some point, irrespective of the actual value at that point. The actual value of a function at a point may not be equal to the limit of the function at that point (as in the case of functions with point discontinuities -- described in another post). The reason such a contorted notion was concocted is that it turns out to be quite useful, in fact fundamental to almost all modern technology. It started with an attempt to find the equation of the tangent to a curve. The slope of the tangent was defined to be the "limit" of the slopes of secant lines as the two points defining the secant lines came close to each other. Note: when the two points that define the secant line touch each other then the slope of the secant line is no longer defined (as the denominator is zero) but the limit (i.e. the slope of the tangent) does exist as one should expect since a tangent is quite a real and tangible concept.
It does take some work to wrap this notion into once head. Some examples will certainly help but I see no point in adding to the excellent examples others have provided. A good calculus text such as that by Tom Apostol will certainly help.
The notion of a limit can be vastly generalised above and beyond what is described above. For instance one could extend the concept of limit to apply to functions of complex numbers. One can even talk of limits of sequences of functions. If one is talking about the limit of a sequence of functions then the limit itself usually is a function (perhaps you are referring to this as a "limit function" ) ?
Let us for the sake of simplicity assume we are talking about functions that take a single real number (as opposed to complex number) as its argument and provides a single real number as its value.
Definition : A real number L is said to be the limit of a real valued function f(x) at a point A, if for every number epsilon > 0, there exits a number delta > 0 such | f(x) - L | < epsilon when ever | x - A | < delta .
What this means is that the limit "L" of a function if it exists, is defined point wise i.e. for each point where the function is defined. The limit may be different at different points. Further this "L" has a special property, roughly stated as follows
1) if you give my any positive number epsilon no matter how small
2) then I can find you a positive number delta such that
3) for all points within a distance delta of A
4) the value of the function is within a distance epsilon of L
5) the value of delta i will give you will depend on the value of epsilon you give me. I will change delta if you change epsilon.
This is a very precise albeit mind bending way to define the notion of the value a function "tends to" at some point, irrespective of the actual value at that point. The actual value of a function at a point may not be equal to the limit of the function at that point (as in the case of functions with point discontinuities -- described in another post). The reason such a contorted notion was concocted is that it turns out to be quite useful, in fact fundamental to almost all modern technology. It started with an attempt to find the equation of the tangent to a curve. The slope of the tangent was defined to be the "limit" of the slopes of secant lines as the two points defining the secant lines came close to each other. Note: when the two points that define the secant line touch each other then the slope of the secant line is no longer defined (as the denominator is zero) but the limit (i.e. the slope of the tangent) does exist as one should expect since a tangent is quite a real and tangible concept.
It does take some work to wrap this notion into once head. Some examples will certainly help but I see no point in adding to the excellent examples others have provided. A good calculus text such as that by Tom Apostol will certainly help.
The notion of a limit can be vastly generalised above and beyond what is described above. For instance one could extend the concept of limit to apply to functions of complex numbers. One can even talk of limits of sequences of functions. If one is talking about the limit of a sequence of functions then the limit itself usually is a function (perhaps you are referring to this as a "limit function" ) ?
