Math 314 Lecture #12 14.2: Limits and Continuity

x 2sin y x2 +2y 2 = lim x→0 x 2sin2 mx x +2m2x2 = lim x→0 sin mx 1+2m2 = 0. We might suspect that the limit exists and is equal to 0. To justify this, we notice that since 0 ≤ x2 x2 +2y2 ≤ 1, we have the inequalities 0 ≤ x 2sin y x 2+2y ≤ sin2 y. The limits of the outer two functions as (x,y) → (0,0) are both 0, and so the Squeeze ... ................
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