Find the limit, if it exists. Step 1 xy lim (x, y) (0, 0) x...

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Find the limit, if it exists. Step 1 xy lim (x, y) (0, 0) √ x² + y² To examine xy lim (x,y) → (0, 0) √ x² + On this path, all points have y = 0 xy lim (x, y) (0, 0) √x² + y² Step 2 Since y = 0 for all points on this path, then we have When x is positive, we have xy lim (x, y) (0, 0) √ x² + y² first approach (0, 0) along the x-axis. When x is negative, we have xy lim (x, y) (0, 0) √ x² + y² = lim X→ 0 Step 3 Similarly, approaching along the y-axis yields a limit equal to 0. Since these two limits are the same, we will examine another approach path. Approach (0, 0) along the curve y = x². = lim 0 + X X. 0 X y = 0 = lim x x → 0 3 3 √ x² + 0 x² + x4 x² + x4 lim x → 0 lim X→ 0- 0 x + x² x² x+x³ X X Find the limit, if it exists. Step 1 xy lim (x, y) (0, 0) √ x² + y² To examine xy lim (x,y) → (0, 0) √ x² + On this path, all points have y = 0 xy lim (x, y) (0, 0) √x² + y² Step 2 Since y = 0 for all points on this path, then we have When x is positive, we have xy lim (x, y) (0, 0) √ x² + y² first approach (0, 0) along the x-axis. When x is negative, we have xy lim (x, y) (0, 0) √ x² + y² = lim X→ 0 Step 3 Similarly, approaching along the y-axis yields a limit equal to 0. Since these two limits are the same, we will examine another approach path. Approach (0, 0) along the curve y = x². = lim 0 + X X. 0 X y = 0 = lim x x → 0 3 3 √ x² + 0 x² + x4 x² + x4 lim x → 0 lim X→ 0- 0 x + x² x² x+x³ X X

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