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





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.



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© A2, September 20, 2003
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