Difference between revisions of "Limit"

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(Properties of a Limit)
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A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.
 
A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.
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Limits are written in the following form:
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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''.
  
 
==Intuitive Definition==
 
==Intuitive Definition==
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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>.   
 
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>.   
  
Indeed for this function, we define the limit as
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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.
<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|400px]]
 
  
 
For example, given  
 
For example, given  
  
<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\  \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math>  
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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)
 
 
  
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[[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).
 
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).
  
 
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.   
 
For cases in general, the expression below satisfies our needs.
 
<math> \lim_{x \to a}f(x) = L </math>
 
 
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''.
 
 
  
 
==Rigorous Definition of a Limit==
 
==Rigorous Definition of a Limit==

Revision as of 10:31, 18 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.

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,  \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 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)\, 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.

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