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Limits and Continuity in Reality
Transcript of Limits and Continuity in Reality
calculus is used in modeling engineering problems, which are often stated as ordinary or partial differential equations. The solutions to these equations can be whole families of curves, from which the engineer must select a physically meaningful particular solution. A continuity condition is often imposed as one of a number of possible boundary conditions to select the physically correct solution.
A function is to be said continuous at x=a if there is no interruption in the graph of f(x) at a. Its graph is unbroken at a, and there is no hole, jump, or gap.
Limit of convergent matrix
Convergence of random variables
In making calculations, engineers will approximate a function using small differences in the a function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals.
As an example, we could have a chemical reaction in a beaker start with two chemicals that form a new compound over time. The amount of the new compound is the limit of a function as time approaches infinity.
In the study Calculus, the first important concept of idea that must be introduced is the concept of limits. The limit of a function is the cornerstone of both differential and integral calculus. Derivatives depends on the notion of the limit of a function
Limits are also used as real-life approximations to calculating derivatives. Real-life limits are used any time you have some type of real-world application approach a steady-state solution.
Real life Limits
Similarly, if you drop an ice cube in a glass of warm water and measure the temperature with time, the temperature eventually approaches the room temperature where the glass is stored. Measuring the temperature is a limit again as time approaches infinity.
For example, when designing the engine of a new car, an engineer may model the gasoline through the car's engine with small intervals called a mesh, since the geometry of the engine is too complicated to get exactly with simply functions such as polynomials. These approximations always use limits.
Limits and Continuity in Reality
Functions that are continuous at every number in a given interval are sometimes thought of as function whose graphs can be sketched without lifting the ballpen from the paper; that is, there is no break in the graph.
Real life Continuities
The field of point-set topology defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function. We can have much weirder functions that don't have numbers as inputs or outputs and yet are still continuous.
Continuous functions, believe it or not, are all sorts of useful. For one thing, they're the secret behind digital recording, including CDs and DVDs.
A brief explanation:
Suppose you want to use a digital recording device to record yourself singing in the shower. The song comes out as a continuous function. The digital recording device can't record what you sound like several times a second.
Since the song is a continuous function and continuous functions are nice, the several-times-a-second recording contains enough information for a computer to reproduce more-or-less what you sounded like the whole time you were singing, if little bits were recorded frequently enough and carefully enough, the reproduction will sound just like you.