Difference between revisions of "Limit"

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(New page: A limit is the behavior of ==Intuitive Definition== We know that a function assigns values to outputs with respect to inputs. As given inputs get closer and closer to a specific value, ...)
 
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A limit is the behavior of  
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A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.
  
 
==Intuitive Definition==
 
==Intuitive Definition==
We know that a function assigns values to outputs with respect to inputs.  As given inputs get closer and closer to a specific value, the function's output get closer and closer to a certain separate value.  We define this value that the function's output approaches as the ''limit''. 
 
 
 
We can examine the limit of a simple continuous function, <math> f(x)=x^2 \,</math>.
 
We can examine the limit of a simple continuous function, <math> f(x)=x^2 \,</math>.
  
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<math> \lim_{x \to a}f(x) = f(a)  \,</math>.  But this is a special case of limit, in the majority of limits cannot be solved in this manner.
 
<math> \lim_{x \to a}f(x) = f(a)  \,</math>.  But this is a special case of limit, in the majority of limits cannot be solved in this manner.
  
[[Image:SimpleParabola.png|right|350px]]
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[[Image:SimpleParabola.png|400px]]
  
 
For example, given  
 
For example, given  
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The limit of <math>f(x)\,</math> as ''x'' approaches 0 is 0 (just as in <math>f(x)\,</math>), but <math>\lim_{x\to 0}f(x)\neq f(0)</math>; <math>f(x)</math> is not continuous at <math>x = 0</math> (as shown on the right).
 
The limit of <math>f(x)\,</math> as ''x'' approaches 0 is 0 (just as in <math>f(x)\,</math>), but <math>\lim_{x\to 0}f(x)\neq f(0)</math>; <math>f(x)</math> is not continuous at <math>x = 0</math> (as shown on the right).
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In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values. 
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For cases in general, the expression below satisfies our needs.
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<math> \lim_{x \to a}f(x) = L </math>
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The expression above states that when <math>x\,</math> approaches arbitrarily close to <math>a\,</math>, the function <math>f(x)\,</math>  becomes arbitrarily close to the value <math>L\,</math>, which is called the ''limit''.
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==Rigorous Definition of a Limit==
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This definition is more appropriate for 2nd year calculus students and higher.
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The original statement <math>\lim_{x \to a}f(x) = L </math> now means that given any <math>\varepsilon > 0 </math>, a <math> \delta > 0 \,</math> exists such that if <math>0 < |x-a| < \delta \,</math>, then <math>|f(x)-L|< \varepsilon  </math>.
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[[Image:Limit.png|right|thumb|300px]]
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We use the definition as follows to prove for the function <math>f(x) = 3x-1 \,</math>, <math>\lim_{x \to 2}f(x) = 5 </math>
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Given any <math>\delta\,</math>, we choose a <math>\varepsilon\,</math> such that <math>\delta \leq \varepsilon/3</math>
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If <math>|x-2|<\delta\,</math>, we can derive <math>|f(x)-5|<3\delta\,</math>.
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<math>3|x-2|<3\delta\,</math>
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<math>|3x-6|<3\delta\,</math>
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<math>|f(x)-5|<3\delta\,</math>
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<math>3\delta \leq 3*\varepsilon/3 = \varepsilon\,</math>
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Thus <math>\lim_{x \to 2}f(x) = 5 </math>
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==Properties of a Limit==
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The limit of a sum of two functions is equal to the sum of the limits of the functions.
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<math> \lim (f(x)+g(x)) = \lim f(x) + \lim g(x) </math>
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The limit of a difference between two functions is equal to the difference between the limits of the functions.
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<math> \lim (f(x)-g(x)) = \lim f(x) - \lim g(x) </math>
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The limit of a product of two functions is equal to the product of the limits of the functions.
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<math> \lim (f(x)*g(x)) = \lim f(x) * \lim g(x) </math>
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The limit of a quotient of two functions is equal to the quotient of the limits of the functions (assuming a non-zero denominator).
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<math> \lim (f(x)/g(x)) = \lim f(x) / \lim g(x) </math>

Revision as of 10:10, 18 June 2009

A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.

Intuitive Definition

We can examine the limit of a simple continuous function,  f(x)=x^2 \,.

The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, f(0)=0.

Indeed for this function, we define the limit as  \lim_{x \to a}f(x) = f(a)  \,. But this is a special case of limit, in the majority of limits cannot be solved in this manner.

SimpleParabola.png

For example, given

f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.


The limit of f(x)\, as x approaches 0 is 0 (just as in f(x)\,), but \lim_{x\to 0}f(x)\neq f(0); f(x) is not continuous at x = 0 (as shown on the right).

In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.

For cases in general, the expression below satisfies our needs.  \lim_{x \to a}f(x) = L

The expression above states that when x\, approaches arbitrarily close to a\,, the function f(x)\, becomes arbitrarily close to the value L\,, which is called the limit.


Rigorous Definition of a Limit

This definition is more appropriate for 2nd year calculus students and higher.

The original statement \lim_{x \to a}f(x) = L now means that given any \varepsilon > 0 , a  \delta > 0 \, exists such that if 0 < |x-a| < \delta \,, then |f(x)-L|< \varepsilon  .

Limit.png


We use the definition as follows to prove for the function f(x) = 3x-1 \,, \lim_{x \to 2}f(x) = 5

Given any \delta\,, we choose a \varepsilon\, such that \delta \leq \varepsilon/3

If |x-2|<\delta\,, we can derive |f(x)-5|<3\delta\,.

3|x-2|<3\delta\,

|3x-6|<3\delta\,

|f(x)-5|<3\delta\,

3\delta \leq 3*\varepsilon/3 = \varepsilon\,

Thus \lim_{x \to 2}f(x) = 5

Properties of a Limit

The limit of a sum of two functions is equal to the sum of the limits of the functions.

 \lim (f(x)+g(x)) = \lim f(x) + \lim g(x)

The limit of a difference between two functions is equal to the difference between the limits of the functions.

 \lim (f(x)-g(x)) = \lim f(x) - \lim g(x)

The limit of a product of two functions is equal to the product of the limits of the functions.

 \lim (f(x)*g(x)) = \lim f(x) * \lim g(x)

The limit of a quotient of two functions is equal to the quotient of the limits of the functions (assuming a non-zero denominator).

 \lim (f(x)/g(x)) = \lim f(x) / \lim g(x)