Limit and Continuity

Limit and Continuity

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 limit and direct limit in category theory.

In formulas, a limit is usually written as

 \lim_{n \to c}f(n) = L

and is read as “the limit of f of n as n approaches c equals L“. Here “lim” indicates limit, and the fact that function f(n) approaches the limit L as n approaches c is represented by the right arrow (→), as in

f(n) \to L \ . 
For example, if

 f(x) = \frac{x^2 - 1}{x - 1}

then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2

In other words,  \lim_{x \to 1} \frac{x^2-1}{x-1} = 2

This can also be calculated algebraically, as \frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1 for all real numbers x ≠ 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus \lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2

That means if you want to solve a Limit based question, you have to Factorise given function, then put the value.

You will get your answer…..



Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus, and integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that elementary algebra alone cannot

Arithmatics, Mathematics


Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory

Totally dependent on the operation of Addition,Subtraction, Multiplication and Division.


Astronomy, Knowledge



An extraordinary ribbon of hot gas trailing behind a galaxy like a tail has been discovered using data from NASA’s Chandra X-ray Observatory. This ribbon, or X-ray tail, is likely due to gas stripped from the galaxy as it moves through a vast cloud of hot intergalactic gas. With a length of at least 250,000 light years, it is likely the largest such tail ever detected. In this new composite image, X-rays from Chandra (blue) have been combined with data in visible light from the Isaac Newton Group of Telescopes (yellow) in the Canary Islands, Spain.

The tail is located in the galaxy cluster Zwicky 8338, which is almost 700 million light years from Earth. The length of the tail is more than twice the diameter of the entire Milky Way galaxy. The tail contains gas at temperatures of about ten million degrees, about twenty million degrees cooler than the intergalactic gas, but still hot enough to glow brightly in X-rays that Chandra can detect.

The researchers think the tail was created as a galaxy known as CGCG254-021, or perhaps a group of galaxies dominated by this large galaxy, plowed through the hot gas in Zwicky 8338. The pressure exerted by this rapid motion caused gas to be stripped away from the galaxy.

In images from Chandra and the NSF’s Karl Jansky Very Large Array (not shown in composite), the galaxy CGCG254-021 appears to be moving towards the bottom of the image with the tail following behind. There is a significant gap between the X-ray tail and the galaxy, the largest ever seen. The significant separation between the galaxy and the tail might be evidence that the gas has been completely stripped off the galaxy.

Astronomers were also able to learn more about the interactions of the system by carefully examining the properties of the galaxy and its tail. The tail has a brighter spot, referred to as its ‘head’. Behind this head is the tail of diffuse X-ray emission. The gas in the head may be cooler and richer in elements heavier than helium than the rest of the tail. In front of the head there are hints of a bow shock, similar to a shock wave formed by a supersonic plane and in front of the bow shock is the galaxy CGCG254-021.

Independent research involving observations at infrared wavelengths indicates that CGCG254-021 has the highest mass of all galaxies in Zwicky 8338. The infrared observations, together with models for how galaxies evolve, also imply that among the galaxies in the cluster, CGCG254-021 had by far the highest rate of stars forming in the recent past. However, there is no evidence for new star formation, possibly because gas has been depleted in forming the tail.

The paper describing these results was published in the November 2015 issue of Astronomy and Astrophysics and is also available online. The authors of the paper are Gerrit Schellenberger and Thomas Reiprich from the University of Bonn in Germany. NASA’s Marshall Space Flight Center in Huntsville, Alabama, manages the Chandra program for NASA’s Science Mission Directorate in Washington. The Smithsonian Astrophysical Observatory in Cambridge, Massachusetts, controls Chandra’s science and flight operations. Swift is managed by NASA’s Goddard Space Flight Center in Greenbelt, Maryland.

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