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Math Help - Computing Limits

  1. #1
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    Computing Limits

    When I am computing a limit where the degree of the numerator is different from the degree of the denominator or vice versa what would I multiply both by in order to solve. Also if there is a square root does it change the what I would multiply by. It's easy to solve the simple limits however I am looking for a set of rules to follow when I am trying to solve the more complex problems.

    Thanks
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    Quote Originally Posted by qbkr21 View Post
    When I am computing a limit where the degree of the numerator is different from the degree of the denominator or vice versa what would I multiply both by in order to solve. Also if there is a square root does it change the what I would multiply by. It's easy to solve the simple limits however I am looking for a set of rules to follow when I am trying to solve the more complex problems.

    Thanks
    Degree of the numerator is different from the denominator...
    Square root changes...

    There are too many possibilities here. Narrow it down or give an example of what you want!

    -Dan
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    2 Examples would be real nice

    1.
    Degree of the numerator is different from the denominator...

    2.
    Square root in the denominator
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by qbkr21 View Post
    2 Examples would be real nice

    1.
    Degree of the numerator is different from the denominator...

    2.
    Square root in the denominator
    I'm still not sure of what you mean. Let's try this:
    \lim_{x \to 0} \frac{x^2 + 2x + 5}{x + 1} = \frac{0^2 + 2*0 + 5}{0 + 1} = 5
    Is this what you are looking to find out how to solve?

    -Dan
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  5. #5
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    Yes now what did you multiply the numerator and denominator by?

    Not a constant but infinity
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    Quote Originally Posted by qbkr21 View Post
    Yes now what did you multiply the numerator and denominator by?

    Not a constant but infinity
    You do not multiply by anything. Since you can substitute there is no need.
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    Rules? I need some rules to play the game...
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    Quote Originally Posted by qbkr21 View Post
    Rules? I need some rules to play the game...
    Yes.

    Math is a game of rules and no objectives.
    Philosophy is a game with objectives and no rules.
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    Could someone please tell me the rules so that I can solve a hard limit problem that goes to Infinity...
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  10. #10
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    There are no "rules" as such, just methods that have been shown to work. And there is usually more than one way to think about them. So you have to work some problems and find what you are most comfortable with.

    \lim_{x \to \infty} \frac{x^2 - 3x + 5}{x + 10}

    Note that as x goes to infinity only the terms of largest degree will be important. So we can do this in two ways:
    1) Keep only the largest degree terms in the numerator and denominator:
    \lim_{x \to \infty} \frac{x^2 - 3x + 5}{x + 10} \to \lim_{x \to \infty}\frac{x^2}{x} = \lim_{x \to \infty}x \to \infty

    2) We can reduce the order of the numerator and denominator by dividing both by the largest power term, in this case x^2:
    \lim_{x \to \infty}\frac{x^2 - 3x + 5}{x + 10} = \lim_{x \to \infty} \frac{1 - \frac{3}{x} + \frac{5}{x^2}}{\frac{1}{x} + \frac{10}{x^2}}
    Notice that any term \lim_{x \to \infty}\frac{1}{x^n} \to 0, so this becomes:
     \to \frac{1}{0} \to \infty

    These processes can be done using any degree polynomials in the numerator or denominator. Also it does NOT matter that we are taking a limit as x goes to infinity, you can use similar arguments as x approaches any number as well.

    -Dan
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