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Math Help - Limit of multivariable function

  1. #1
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    Limit of multivariable function

    The question asks to find the limit (x,y) -> (0,0) and this is the function:
     \frac{x^2 sin(y)^2}{x^2 + 5y^2}
    so i tried approaching (0,0) from y=cx and i get this:
     \frac{x^2 sin(cx)^2}{x^2(1 + 5c)}
    but im not quite sure where to go from here??
    an explanation would be greatly appreciated
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  2. #2
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    This may not be correct in the slightest so please don't take my word for it

    First of all you made a slight mistake it should be

    \frac{x^2 \sin^2 (cx)}{x^2(1+5c^2)}

    so

    =\frac{\sin^2 (cx)}{1+5c^2} (you can just cancel the x^2's

    =\frac{(cx-\frac{cx^3}{6}+ \ldots)^2}{1+5c^2} (the taylor expansion of sin is probably unnecessary!)

    hence as x goes to 0 (and so y goes to 0 along the line y=cx) the limit approaches 0 for c \neq 0
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  3. #3
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    thank you for catching my mistake
    just one question: if i had just assumed that since as x approaches 0, sin(0)=0 .. so the limit =0, would that be alright??
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  4. #4
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    yea it would be fine!
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