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In calculus, limits are extremely important because calculus itself is based upon limits. Basically, a limit describes the behavior of a dependant variable when its independent variable takes extreme values.
For example, lets consider this function : y =1/x. As you know, this typical function is not defined at x = 0 because the division by zero is not admitted in real numbers. Therefore we cannot compute the value of y when x = 0. However, we can observe how the function behaves near x = 0 : this is the concept of limit. Lets see how this function behaves when x approaches zero from the right:
limx->0+ 1/x = infinity
You can verify this limit by substituting x with values that approach zero: 1/0.1 = 10; 1/0.000009 = 11,111.11; ... When x takes extremely small values, y takes extremely large values. If we repeat this process forever, we will say that, at the limit, y will have an infinite value. That's what the limit is.
Two of the most important uses of limits in calculus are derivatives/differentials and integrals. For example, the derivative of a function is the limit of the function's ratio of variation of dependant and independent variables as the variation of the independent variable approaches zero:
derivative of f(x) with respect to x = f'(x) = d[f(x)]/dx ...
Also, a definite integral is defined as the limit of the sum of infinitely small elements as the number of elements approaches infinity.
To calculate limits, you must have a good knowledge of the "algebra of infinity" (infinity + infinity = infinity, sqrt(infinity) = infinity, ...).Of course, this is a very basic description of the limit concept; there are many, many cases where the limit is tricky to calculate.
The concept of Functions limits and Continuity leads to define and describe continuity and derivative of the function.
The continuity of a function has practical as well as theoretical importance. We plot graphs by taking the values generated in the laboratory or collected in the field. We connect the plotted points with a smooth and unbroken curve (continuous curve). This continuous curve helps as to estimate the values at the places where we haven't measured. It was developed by Isaac newton and Leibnitz.
Here in this chapter, we will study some standard functions, their graphs, concept of limits and discuss about the continuity of the functions. Throughout this chapter, we denote R as the set of real numbers.
Types of Limit:
Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit.
Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.
We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal.
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Calculus is a branch of mathematics that studies what?
Calculus is the branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit, that is, the notion of tending toward, or approaching, an ultimate value.
What is calculus about?
Calculus is about applying the idea of limits to functions in various ways. For example, the limit of the slope of a curve as the length of the curve approaches zero, or the limit of the area of rectangle as its length goes to zero. Limits are also used in the study of infinite series as in the limit of a function of xas x approaches infinity.
Define The LIMIT as in Calculus?
The term "limit" in calculus describes what is occurring as a line approaches a specific point from either the left or right hand side. Some limits approach infinity while some approach specific points depending on the function given. If the function is a piece-wise function, the limit may not reach a specific value depending on the function given. For a more in-depth definition here is a good link to use: * http://www.math.hmc.edu/calculus/tutorials/limits/
What is calculus in mathematics?
Calculus involves the exploration of limits in mathematics. For example, if you consider a polygon and keep adding a side to it, eventually it will begin to look like a circle but it will never truly be a circle. This is an example of a limit.
What is an example of a calculus limit?
the limit [as x-->5] of the function f(x)=2x is 5 the limit [as x-->infinity] of the function f(x) = 2x is infinity the limit [as x-->infinity] of the function f(x) = 1/x is 0 the limit [as x-->infinity] of the function f(x) = -x is -infinity
What is the history of limits in calculus?
newton and Leibniz were first introduced the concept of limit independently
Does infinity have a limit?
No. That is why it is called "infinity". Infinity is actually not an accepted numerical value in calculus. It is rather a concept. For instance, (infinity) - 1 googleplex = infinity
What is limits in calculus?
In calculus, a limit is a value that a function or sequence approaches as the input values get closer and closer to a particular point or as the sequence progresses to infinity. It is used to define continuity, derivatives, and integrals, among other concepts in calculus. Calculus would not be possible without the concept of limits.
What are the foundation of differential calculus and integral calculus?
The foundation, in both cases, is the concept of limits. Calculus may be said to be the "study of limits". You can apply a lot of calculus in practice without worrying too much about limits; but then we would be talking about practical applications, not about the foundation.
Calculus is a branch of mathematics that studies what?
Calculus is the branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit, that is, the notion of tending toward, or approaching, an ultimate value.
What is calculus about?
Calculus is about applying the idea of limits to functions in various ways. For example, the limit of the slope of a curve as the length of the curve approaches zero, or the limit of the area of rectangle as its length goes to zero. Limits are also used in the study of infinite series as in the limit of a function of xas x approaches infinity.
What is meant by the Law of Infinitesimals?
The Law of Infinitesimals states that as quantities decrease without limit, their effects become negligible or zero. In calculus, this concept is used to define derivatives and integrals, where infinitesimally small changes lead to the foundation of differential and integral calculus.
Why are tangent lines important?
Because it leads to the limit concept which in turn leads to concept of derivative...
Define The LIMIT as in Calculus?
The term "limit" in calculus describes what is occurring as a line approaches a specific point from either the left or right hand side. Some limits approach infinity while some approach specific points depending on the function given. If the function is a piece-wise function, the limit may not reach a specific value depending on the function given. For a more in-depth definition here is a good link to use: * http://www.math.hmc.edu/calculus/tutorials/limits/
What is the definition of fatigue endurance limit and stress ratio?
B.Sc PCM- 1 year (APPLIED CALCULUS) 18.09.2012
What is the meaning of the limit concept in the mathematical context?
It can be difficult to remember all mathematical terms and their meanings. The limit concept is the value that a function or sequence approaches as the input approaches a value.Ê
What is calculus in mathematics?
Calculus involves the exploration of limits in mathematics. For example, if you consider a polygon and keep adding a side to it, eventually it will begin to look like a circle but it will never truly be a circle. This is an example of a limit.
