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Math Help - L'hopital for more than one variable?

  1. #1
    jbd
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    L'hopital for more than one variable?

    Hi I have to compute functions with the basic form: sin(x)sin(y)/(xy) over values including where x is zero and y is non-zero (and vice-versa) - for which I have used L'hopital... but also where both x and y are zero for which I really don't know what to do? Could anyone help me out? Thanks kindly.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by jbd View Post
    Hi I have to compute functions with the basic form: sin(x)sin(y)/(xy) over values including where x is zero and y is non-zero (and vice-versa) - for which I have used L'hopital... but also where both x and y are zero for which I really don't know what to do? Could anyone help me out? Thanks kindly.
    Are you talking about

    \lim_{x\to{0}}\frac{\sin(x)\sin(y)}{xy}?

    If so rewrite it as

    \lim_{x\to{0}}\frac{\sin(x)}{x}\cdot\lim_{x\to{0}}  \frac{\sin(y)}{y}

    Now the left lmiit can be done a million ways to be shown as one and the right is just itself

    \therefore\lim_{x\to{0}}\frac{\sin(x)\sin(y)}{xy}=  \frac{\sin(y)}{y}

    The millions of ways include

    The definition of the derivative at apoint

    its is well known that f'(c)=\lim_{x\to{c}}\frac{f(x)-f(c)}{x-c}

    Thus directly from that we see that

    \lim_{x\to{0}}\frac{\sin(x)}{x}=\lim_{x\to{0}}\fra  c{\sin(x)-\sin(0)}{x-0}=\bigg(\sin(x)\bigg)\bigg|_{x=0}=\cos(0)=1
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  3. #3
    Moo
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    Hello,

    Quote Originally Posted by jbd View Post
    Hi I have to compute functions with the basic form: sin(x)sin(y)/(xy) over values including where x is zero and y is non-zero (and vice-versa) - for which I have used L'hopital... but also where both x and y are zero for which I really don't know what to do? Could anyone help me out? Thanks kindly.
    Were you specifically told to use l'H˘pital's rule ?

    \frac{\sin x \sin y}{xy}=\frac{\sin x}{x} \cdot \frac{\sin y}{y}

    And these are supposed to be known limits
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  4. #4
    jbd
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    I'm happy with the limit sinx/x where x goes to zero but I have both x and y going to zero at the same time... can I just say that sinx/x goes to 1 and siny/y goes to 1 so the limit is 1.1 = 1?
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  5. #5
    jbd
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    I haven't be told to use L'hopital - this is something I've come across writing a program for a physics problem I'm working on for my PhD - it's actually a while since I did any of this stuff for undergrad.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by jbd View Post
    I haven't be told to use L'hopital - this is something I've come across writing a program for a physics problem I'm working on for my PhD - it's actually a while since I did any of this stuff for undergrad.
    Oh, you mean this

    \lim_{(x,y)\to{0}}\frac{\sin(x)\sin(y)}{xy}

    This limit only exists if

    \lim_{x\to{0}}\frac{\sin(x)\sin(y)}{xy}=\lim_{y\to  {0}}\frac{\sin(x)\sin(y)}{xy}
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  7. #7
    Moo
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    Quote Originally Posted by Mathstud28 View Post
    Oh, you mean this

    \lim_{(x,y)\to{0}}\frac{\sin(x)\sin(y)}{xy}

    This limit only exists if

    \lim_{x\to{0}}\frac{\sin(x)\sin(y)}{xy}=\lim_{y\to  {0}}\frac{\sin(x)\sin(y)}{xy}
    Not "only".
    It has to tend to the same value for any direction of x or y with respect to y or x. This is for continuity =)
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  8. #8
    jbd
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    So what is the limit if that condition holds? Sorry for being slow (also sorry for notating better!)
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  9. #9
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    L'Hospital rule

    As far as I know there is no L'Hospital rule for more than one variables.

    However as it was pointed out before this limit is trivial...


    We should use the following identity (which is true if all the limits exists):

    \lim_{(x,y) \rightarrow (x_0, y_0)} f(x, y) g(x, y) = \lim_{(x,y) \rightarrow (x_0, y_0)} f(x, y) \lim_{(x,y) \rightarrow (x_0, y_0)} g(x, y)

    Than we should note:

    \lim_{(x,y) \rightarrow (x_0, y_0)} h(x) = \lim_{x \rightarrow x_0} h(x)
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  10. #10
    jbd
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    Smile

    OK, I'm feeling a lot clearer on this now. Thanks everyone for your help.
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