Difference between revisions of "Limit"

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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''.  
 
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''.  
  
==Intuitive Definition==
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==Informal Definition==
 
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>.
  
The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, <math>f(0)=0</math>.   
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The limit at x=0 of f(x) = x^2 would pretty clearly be 0, since <math>f(0)=0</math>.   
  
 
Indeed for this function, <math> \lim_{x \to a}f(x) = f(a)  \,</math>.  But this is a special case, in the majority of limits cannot be solved in this manner.
 
Indeed for this function, <math> \lim_{x \to a}f(x) = f(a)  \,</math>.  But this is a special case, in the majority of limits cannot be solved in this manner.
  
For example, given  
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For a very different example; given  
  
 
<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below)
 
<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below)
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[[Image:SimpleParabola.png|400px]]
 
[[Image:SimpleParabola.png|400px]]
  
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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The limit of <math>f(x)\,</math> because ''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).
  
 
In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.
 
In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.
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<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below)
 
<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below)
  
==Rigorous Definition of a Limit==
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==Rigorous Definition of Limit==
 
This definition is more appropriate for 2nd year calculus students and higher.
 
This definition is more appropriate for 2nd year calculus students and higher.
  
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}}
 
}}
  
==Properties of a Limit==
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==Properties of Limits==
 
{{hide|1=The limit of a sum of two functions is equal to the sum of the limits of the functions.
 
{{hide|1=The limit of a sum of two functions is equal to the sum of the limits of the functions.
  

Revision as of 12:10, 25 June 2009

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

Limits are written in the following form:

 \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.

Informal Definition

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

The limit at x=0 of f(x) = x^2 would pretty clearly be 0, since f(0)=0.

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

For a very different example; given

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

SimpleParabola.png

The limit of f(x)\, because 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.

Here we look at one such case:

EvaluatingLimitsDiff1.PNG

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

Rigorous Definition of 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 Limits

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 (\frac{f(x)}{g(x)}) = \frac{\lim f(x)}{\lim g(x)}