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Posted by on Saturday, September 18, 2010 at 11:19pm.

Our professor wants us to evaluate the limits analytically without using a table or a graph, and if it doesn't exist we must describe the behavor near the limit point. I'm not sure how to evaluate each side of a limit separately without looking at a table or graph. Here is one of the problems:

The limit of (sqrt(2-x)-2)/x as x approaches 0.

  • Calculus - , Saturday, September 18, 2010 at 11:37pm

    Didn't I show you this one? Rationalize the numerator;

    lim (sqrt(2-x)-2)(sqrt(2-x)+2)/(x(sqrt(2-x)+2)

    which equals
    lim ((2-x-4)/(x(sqrt(x-2)+2)

    which equals
    lim (x-4)/(x(sqrt(x-2)+2)

    as x approaches zero
    -4/zero which does not exist.

  • Calculus - , Saturday, September 18, 2010 at 11:44pm

    But now if it does not exist we have to go back and look at the limit as the function approaches from the left and from the right and describe the behavoir of the function for example if it approaches infinity or negative infinity. I don't know how to do that analytically without using a graph or a chart.

  • Calculus - , Saturday, September 18, 2010 at 11:44pm

    I'm not sure if you learned about L'hospital rule
    http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx

    (sqrt(2-x)-2)/x

    rewrite:
    [(2-x)^(-1/2)-2]/x

    perform L'hospital's by taking the derivative of numerator and denominator individually:
    [(1/2)(2-x)^(-1/2)]/1

    now take the limit.

  • Calculus - , Saturday, September 18, 2010 at 11:46pm

    opps, forgot about chain rule! be sure to take the derivative of the inside (2-x)!

    [(-1/2)(2-x)^(-1/2)]/1

  • Calculus - , Sunday, September 19, 2010 at 12:03am

    look at the left. Let x approach from the left (x is negative)
    lim (2-x)-2)/x means the denominator is negative, the numberator is positive, so the limit is negative.
    Now, from the right, x is positive
    lim (2-x)-2)/x numerator is negative, denominator is positive, lim is negative.

  • Calculus - , Sunday, September 19, 2010 at 12:06am

    We don't start derivatives until the next chapter. I'm just don't understand how you can determine each side of a limit separately without using a graph or chart.

  • Calculus - , Sunday, September 19, 2010 at 12:09am

    Anna,
    can you double check if the question is not:
    "The limit of (sqrt(4-x)-2)/x as x approaches 0"

    The answer would be -1/4 using Mr. Pursley's approach.

  • Calculus - , Sunday, September 19, 2010 at 12:19am

    no its a 2.

  • Calculus - , Sunday, September 19, 2010 at 12:22am

    Way back in the stone age when I taught math, I gave
    my students the following procedure when doing limits.
    Subthe approach value in the given expression

    1. If you get a real number as an answer, that's it.
    That is your answer, go on to the next question.
    2. If you get a/0 , where a ≠ 0, then the limit is undefined.
    3. If you get 0/0, you got yourself a real limit question. Try rationalizing, factoring, substitution or other clever math procedures.

    in your case, unless there is a typo like MathMate suspects, you would get (√2 - 2)/0 which is undefined.

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