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cheatsheet - Limit A R f A R c R The limit of f at c exists...

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Limit A R , f : A R , c R . The limit of f at c exists provided 1. a, b R , a < c < b, 3 [( a, c ) ( c, b )] A 2. L R 3 ∀ > 0, δ > 0 3 if 0 < | x - c | < δ , then | f ( x ) - L | < . limit at infinity: A R , f : A R , then the limit as x goes to infinity of f exists provided 1. a R 3 ( a, ) A . 2. L R 3 ∀ > 0 x o R 3 if x > x o then | f ( x ) - L | < . limit in higher dimensions: D R n , f : D R m , and p o R n . The limit of f at p o exists provided that both 1. p o is a cluster point of D . 2. L R m 3 ∀ > 0 δ > 0 3 if p D and 0 < | p - p o | < δ then | f ( p ) - L | < . Thm: p o D R n and f : D R m . Then 1. If p o is an isolated point of D then f is continuous at p o . 2. If p o is a cluster point of D then f is continuous at p o iff lim p po f ( p ) = f ( p o ). Derivative A R , f : A R , z A . f is differentiable at z provided that 1. z A . 2. a, b 3 a < 0 < b , and if a < h < 0 or 0 < h < b then z + h A . 3. L 3 ∀ > 0 δ > 0 3 if 0 < | h | < δ then | f ( z + h ) - f ( z ) h - L | < . Basically, lim h 0 f ( z + h ) - f ( z ) h exists and equals L . Derivative in higher dimensions: S R n , f : S R , and p o is inte- rior point of S . Then f is differentiable at p o provided V R n such that lim Δ p ¯ 0 f ( po p ) - f ( po ) - V · Δ p | Δ p | = 0. Note: if such a V exists, it is equal to the gradient ( f 1 , f 2 , . . . , f n ). Denoted Df ( p o ). How to prove/disprove differentiability: All you have to do is prove that the gradient (vector of partial derivatives) is the derivative or that it is not the derivative. If the partial derivatives don’t exist, then obviously the derivative doesn’t exist either. Thm: S R n is open, p o S and f : S R is C 1 . Then the derivative Df ( p o ) exists. Thm: If S R n , f : S R and p o is an interior point of S . If f is differentiable at p o , then f is continuous at p o . Thm: If S R n and f : S R . If f is differentiable at p o then β R n with | β | = 1, D B f ( p o ) exists and equals Df ( p o ) · β . Derivative in highest dimensions: S R n , f : S R m , and p o is interior point of S . Then f is differentiable at p o provided m × n matrix B such that lim Δ p ¯ 0 f ( po p ) - f ( po ) - B Δ p | Δ p | = ¯ 0 R m . B is the Jacobian, with partial derivatives of f 1 in first row, of f 2 in second row, etc. Directional derivative If D R n , f : D R and p o is interior point of D . β R n and | β | = 1. The directional derivative of f at p o in the direction β is lim t 0 f ( po + ) - f ( po ) t , provided the limit exists. Denoted ( D β f )( p o ).
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