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Announcement for GDB Topic | Dated: Feb 14, 14 |
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The following is the topic for GDB
“Discuss at least three mathematical fields where the concept of “LIMIT’ is used to solve the problems”
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Some fields of Mathematics where LIMITS are applicable are as follows:
1 Calculus
2 Statistics
3 Numerical Analysis
Study about different fields and choose the 3 fields u like and briefly write your views.
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One of the use of limit:-
Limit is used for checking continuty of a function at a specific point, where you come to know whether function is continous at specific point or not.
Limit in terms of Math
Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer! We call that the limit of a function.
In formulas, a limit is usually denoted "lim" as in limn → c(an) = L, and the fact of approaching a limit is represented by the right arrow (→) as in an → L.
Limit is used in many fields of mathematics some are as follows
1 Calculus
2 Statistics
3 Numerical Analysis
In calculus we see there is a so much use of limits involving as follows
Finding instantaneous rate of change of a function.
Instantaneous rate of change
Solving area problems.
In mathematics, the limit of a function is a fundamental concept in calculus andanalysis concerning the behavior of that function near a particular input.
The notion of a limit has many applications in modern calculus. In particular, the many definitions ofcontunityemploy the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of thederivative: in the calculus of one variable, this is the limiting value of the slope of secant linesto the graph of a function.
Limit is used in Statistics
The unexpected appearance of a normal distribution from a population distribution that is skewed (even quite heavily skewed) has some very important applications in statistical practice. Many practices in statistics, such as those involving hypothesesor confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statistics course is that the populations that we work with are normally distributed.
The assumption that data is from a normal distribution simplifies matters, but seems a little unrealistic. Just a little work with some real-world data shows that , skewness, multiple peaks and asymmetry show up quite routinely. We can get around the problem of data from a population that is not normal. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.
Limits are often necessarily in numerical analysis.
A very simple way of calculating numerically would be to approach the limiting by smaller and smaller steps, and just stop whenever the accuracy of the obtained value is high enough.
For example, the following would suffice to calculate numerically (I'm doing it in pseudo-Python here):
x = 1
dx = 0.1
dlim = 1
lim = 0
prev = 0
while abs(dlim) > 1e-6:
if x<=dx: dx *= 0.1
x -= dx
lim = sin(x)/x
dlim = lim-prev
This algorithm will take smaller and smaller steps and approach from above, until the obtained limit value changes by less than the required accuracy. Of course, in this case you could really start with and , since we know that nothing interesting happens before very close to , where Python sooner or later will return NaN
or Inf
because we're dividing by zero...
Limits is used in calculas
In mathematics, the limit of a function is a fundamental concept in calculus andanalysis concerning the behavior of that function near a particular input.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an outputf(x) to every input x. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer top. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to pare taken to outputs that stay a fixed distance apart, we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions ofcontunityemploy the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of thederivative: in the calculus of one variable, this is the limiting value of the slope of secant linesto the graph of a function.
Limit is used in Statistics
The unexpected appearance of a normal distribution from a population distribution that is skewed (even quite heavily skewed) has some very important applications in statistical practice. Many practices in statistics, such as those involving hypothesesor confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statistics course is that the populations that we work with are normally distributed.
The assumption that data is from a normal distribution simplifies matters, but seems a little unrealistic. Just a little work with some real-world data shows that , skewness, multiple peaks and asymmetry show up quite routinely. We can get around the problem of data from a population that is not normal. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.
Thus, even though we might not know the shape of the distribution where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal. Of course, in order for the conclusions of the theorem to hold, we do need a sample size that is large enough. Exploratory data analysis can help us to determine how large of a sample is necessary for a given situation.
Limits are often necessarily in numerical analysis.
A very simple way of calculating numerically would be to approach the limiting by smaller and smaller steps, and just stop whenever the accuracy of the obtained value is high enough.
For example, the following would suffice to calculate numerically (I'm doing it in pseudo-Python here):
x = 1
dx = 0.1
dlim = 1
lim = 0
prev = 0
while abs(dlim) > 1e-6:
if x<=dx: dx *= 0.1
x -= dx
lim = sin(x)/x
dlim = lim-prev
This algorithm will take smaller and smaller steps and approach from above, until the obtained limit value changes by less than the required accuracy. Of course, in this case you could really start with and , since we know that nothing interesting happens before very close to , where Python sooner or later will return NaN
or Inf
because we're dividing by zero...
Note: I thought this up from the top of my head, so it might not be an optimal approach - especially not for an arbitrary, unknown function (which you can evaluate, but you don't know how it behaves...). Also, I cannot really see a use case scenario when you'd need to do this.
In my view some fields of Mathematics where LIMITS are applicable are as follows:
1 Calculus
2 Statistics
3 Numerical Analysis
Study about different fields and choose the 3 fields u like and briefly write your views.
In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limitand direct limit in category theory.
In formulas, a limit is usually denoted "lim" as in limn → c(an) = L, and the fact of approaching a limit is represented by the right arrow (→) as in an → L.
"f(x) gets close to some limit as x gets close to some value"
If we call the Limit "L", and the value that x gets close to "a" we can say
"f(x) gets close to L as x gets close to a"
Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer! We call that the limit of a function.
Limit in Statistics
The normal distribution is used to help measure the accuracy of many statistics, including the sample mean, using an important result called the Central Limit Theorem. This theorem gives you the ability to measure how much the means of various samples will vary, without having to take any other sample means to compare it with. By taking this variability into account, you can use your data to answer questions about a population, such as “What’s the mean household income for the whole U.S.?”; or “This report said 75% of all gift cards go unused; is that really true?” (These two particular analyses are made possible by applications of the Central Limit Theorem called confidence intervalsand hypothesis tests, respectively.)
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