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Engineering Calculus Notes 232

Engineering Calculus Notes 232 - harder to think about than...

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220 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Actually, the situation is even worse: if we consider another function defined on the plane except the origin by f ( x,y ) = x 2 y x 4 + y 2 , ( x,y ) negationslash = (0 , 0) then along a line y = mx through the origin, we have the values f ( x,mx ) = ( x 2 )( mx ) x 4 + ( mx ) 2 = mx 3 x 2 ( x 2 + m 2 ) = mx x 2 + m 2 0 as −→ x −→ 0 . 3 Thus, the limit along every line through the origin exists and equals zero . We might conclude that the function converges to zero as −→ x goes to −→ 0 . However, along the parabola y = mx 2 we see a different behavior: f ( x,mx 2 ) = ( x ) 2 ( mx 2 ) x 4 + ( mx 2 ) 2 = m 1 + m 2 so the limit along a parabola depends on which parabola we use to approach the origin. In fact, we really need to require that the limit of the function along every curve through the origin is the same. This is even
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Unformatted text preview: harder to think about than looking at every sequence converging to −→ 0 . The deFnition of limits in terms of δ ’s and ε ’s, which we downplayed in the context of single variable calculus, is a much more useful tool in the context of functions of several variables. Remark 3.1.6. ( ε-δ Defnition oF limit:) ±or a Function f ( −→ x ) defned on a set D ⊂ R 3 with −→ x an accumulation point oF D , the Following conditions are equivalent: 1. ±or every sequence { −→ x k } oF points in D distinct From −→ x , f ( −→ x k ) → L ; 3 The preceding argument assumes m n = 0. What happens if m = 0?...
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