Unformatted text preview: xc we can write, for all x ∈ A with x 6 = c , g ( f ( x ))g ( f ( c )) xc = h ( f ( x )) f ( x )f ( c )) xc . Now f ( x ) is continuous at c because it is diﬀerentiable there, g ( y ) is continuous at y = f ( c ) for the same reason, so by Theorem 4.3.9 h ( f ( x )) is continuous at x = c : lim x → c h ( f ( x )) = h ( f ( c )) = g ( f ( c )). We can now take the limit as x → c , using the Algebraic Limit Theorem (Corollary 4.2.4): lim x → c g ( f ( x ))g ( f ( c )) xc = lim x → c h ( f ( x )) lim x → c f ( x )f ( c )) xc = g ( f ( c )) f ( c ) . This completes the proof....
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This note was uploaded on 11/07/2009 for the course MATH 3224 at Virginia Tech.
 '08
 KBHANNSGEN
 Calculus, Addition

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