Limit Function - Take the limit as x approaches ...



sketchpad sketchpad
Use Dynamic
Analytic Geometry Material on


limit

  • Take a limit by dragging x and seeing where f(x) is going
  • Take a limit by epsilon-delta
  • Take a limit by examining f(x) above and below x
  • Take a limit by the slope of the secant
  • Take a limit at negative or positive infinity
  • Examine the example at the bottom of this page


Limit is a function.
and
Reading Limit Notation

      A function is a really dependable rule.

      The argument is the thing on which (or with which) the function is operated or performed. In the limit expression below, most would say the argument is the function (x+5)/(x+2). The limiting constant, 2, is the "unstated argument."

     

      See an animation on how to read limit notation.



Use division to transform the expression for easy graphing.

      The function f(x) = (x+5)/(x+2) can be easily seen to have

  • a zero at - 5
    because setting the numerator to zero and solving
    x+5 = 0, results in x = - 5.
  • a vertical asymptote at x = - 2
    because setting the denominator to zero and solving
    x+2=0, results in x = - 2, so the function is undefined at -2.
  • a y-intercept at 5/2
    because replacing x with 0 results in (0+5)/(0+2) or 5/2.

 

      More than that may not be easy to see, but, a little bit of long division makes the rational expression look more like a function that is easy to graph.   Click here to see an animation on the division.



Think APPROACH to take a limit.

      For continuous (and some other) functions, taking a limit requires one simply to approach, get closer and closer, to evaluate the limit.

      See an animation.

      Look at the graph of the function then take a limit graphically. Click on the expression to view the answer.

     

      BY DEFINITION one must approach the limit above and below and these values must be equal for a limit to be evaluated.



Approach the limit from above.

      BY DEFINITION one must approach the limit above and below and these values must be equal for a limit to be evaluated.

      Look at the graph. Approach the limit from above and check the answer by clicking on the expression.

 



Approach the limit from below.

      BY DEFINITION one must approach the limit above and below and these values must be equal for a limit to be evaluated.

      Look at the graph. Approach the limit from below and check the answer by clicking on the expression.

 



Take a limit.

      BY DEFINITION one must approach the limit above and below and these values must be equal for a limit to be evaluated.

      Look at the graph. State the limit and check the answer by clicking on the expression.

 



Take a limit at infinity.

      Look at the graph. Take a limit at negative infinity. Check the answer.



When A Limit Does Not Exist

      Sometimes a limit does not exist. Click on the image to enlarge it.



sketchpad sketchpad
Epsilon-Delta Definition of Limit
limit   -- take a limit dynamically by the delta-epsilon method


      This is not a textbook. No one else that I know presents this topic this way. Here more time is spent on basic vocabulary (particularly if it can be supported by vocabulary already used on this site) and images (both dynamic and static). The limit does this dynamically.

      Here simpler vocabulary is used, believing Isaac Newton and Gottfried Wilhelm Leibniz would not mind. This is a list of vocabulary for one who has forgotten how or never learned how to "speak math."

    Background Vocabulary and Symbols
  • x   - a variable, a number whose value may change within a certain set of numbers, a domain, or which represents any number. Here we use it to represent a number in a domain of numbers.
  • a   - a constant, a number whose value does not change in this instance. This value of a is in the domain of x, a ∈ x.
  • |x - a|   - is the difference between x and a, or, the distance from x to a.
        |x| is the symbol for absolute value of x, the distance x is from 0.
        |x| is also |x - 0| even though this is not often taught.
        |x - 3| is distance from 3 and this is used often in problems like "Graph: y = |x - 3|," but almost never spoken in words.
        Consider the notation |x - a| as number or consider it as a distance from a with work on Exploring Functions: The Distance from Zero and Other Graphs
        Recall that |x - a| is the difference between x and a, or, the distance from x to a.
        If x were equal to a then
        |x - a| would be
        |a - a| or 0. But, x needs only to approach a, not equal a. So, for the task here, since x does not equal a,
        |x - a| > 0, or,
        0 < |x - a|

  • f(x)   - a function with the variable x.
  • f(a)   - the value of the function f(x) when x is equal to the constant a.

      Before reading further, look at and examine the epsilon-delta sheet in the Sketch Pad. It is the same sketchpad that is listed as "limit" often on this page. The goal is to recognize the symbols you know and to become ready to learn the meaning of new symbols.

    Limit Vocabulary and Symbols
  • L   - the limit, f(a). L is the value of the function when x is equal to a, f(a), if the limit exists.
  • epsilon   - error, epsilon is also a positive number, or a length. Here both a positive number and a length are used.

       ε is the Greek letter e. Here the word epsilon, the name of the Greek letter, is used. But that still doesn't explain what error is.

        A more detailed description of error is necessary. This error is how far "off" from L, the limit, one will accept. It is like an error in statistics. Take the just stated error link and scroll up just a bit to see the image that is printed just above the definition, then go back to the spreadsheet picture and examine error there with the epsilon slide and step 4.
       | f(x) - L | < ε
       | f(x) - L | < epsilon, the distance the function is from the limit is less than the chosen error,
       | f(x) - L | < epsilon, the difference between the f(x) function value and the actual f(a) limit is less than the chosen error, and limits are controlled by the error so the error is not 0, so,
       0 < | f(x) - L | < ε
       0 < | f(x) - L | < epsilon

  • delta   - a length or distance or number greater than the distance from x to a, any one of an infinite number of possible lengths, possible positive numbers

       δ is the Greek letter d. Here the word delta, the name of the Greek letter, is used.
       |x - a| < δ
       |x - a|< delta, the distance x is from a is less than delta. Since x does not equal a,
       0 < |x - a| < δ
       0 < |x - a| < delta

        Go back to the sketchpad and examine delta with the delta slide and step 5 then see if the definition below on the left return and the image below on the right make sense.

    See if the definition below on the left return and the image below on the right make sense.

    Click on the images to enlarge them. Play on the sketchpad to clarify the definition.

 


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