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Math Help - Using the delta-epsilon definition of the limit -- solve this problem, please!

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    Post Using the delta-epsilon definition of the limit -- solve this problem, please!

    Show that there exists a d>0, such that f has a limit at (0,0). Find the value of d.
    f(x,y) = (x + y)/(x2 + 1) ; e = 0.01
    Answer: d = 0.005
    (Problem number 63, exercise, 12.2 from Calculus (9th edition by Thomas and Finney.)
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    Quote Originally Posted by taureau20 View Post
    Show that there exists a d>0, such that f has a limit at (0,0). Find the value of d.
    f(x,y) = (x + y)/(x2 + 1) ; e = 0.01
    Answer: d = 0.005
    (Problem number 63, exercise, 12.2 from Calculus (9th edition by Thomas and Finney.)
    We will show the limit is zero.

    \left| \frac{x+y}{x^2+1} - 0 \right| = \frac{|x+y|}{x^2+1} \leq |x+y| \leq |x|+|y|

    Now if 0 < \sqrt{x^2+y^2} < \delta \implies |x|,|y|<\delta

    Therefore |x|+|y| < 2\delta.

    Thus, choose \delta = \frac{\epsilon}{2}.
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