Sheet (4) solution - Math 103 Class Work Sheet (4) Solution...

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1 Math 103 Class Work Sheet (4) Solution [1] (i) 3 ) ( x x f , 3 ) ( ) ( h x h x f 2 2 2 2 2 2 0 2 2 0 3 3 0 0 3 ] ) ( ) [( lim ) ( ' ] ) ( ) ][( ) [( lim ) ( lim ) ( ) ( lim ) ( ' x x x x h x h x x h x h x f h x h x x h x x h x h x h x h x f h x f x f h h h h (ii) 1 ) ( , 1 ) ( h x h x f x x f 1 2 1 1 1 1 ] 1 1 [ lim ] 1 1 [ ) 1 ( ) 1 ( lim ) ( ' 1 1 1 1 . 1 1 lim ) ( ) ( lim ) ( ' 0 0 0 0 x x x x h x h h x h x h x h x x f x h x x h x h x h x h x f h x f x f h h h h (iii) h x h x f x x f 1 ) ( , 1 ) ( 2 0 0 0 0 0 1 ) ( 1 lim ) ( ' ) ( lim ) ( ) ( lim ) ( ) ( . 1 1 lim ) ( ) ( lim ) ( ' x h x x x f h x xh h h x xh h x x h x x h x x h x h x h x f h x f x f h h h h h (iv) h x h x f x x f 1 ) ( , 1 ) ( x x x x x x h x x h x x h h h x x h x x h h x x x f h x x h x x h x x h h x x h x x h x x h x h x h x f h x f x f h h h h h 2 1 ] [ 1 ) .( lim ) .( ) ( lim ) ( ' ) ( ) ( . ) ( lim . 1 1 lim ) ( ) ( lim ) ( ' 0 0 0 0 0 (v) ) cos( ) ( , cos ) ( h x h x f x x f x x x h x h x x f h x x x h x h x h x f h x f x f h h h h h sin ) 1 .( sin ) 0 .( cos sinh . sin lim 1 cosh . cos lim ) ( ' cos sinh . sin cosh . cos lim cos ) cos( lim ) ( ) ( lim ) ( ' 0 0 0 0 0 Note that: 0 0 . 1 2 sinh . 2 2 sin lim 2 sin . 2 lim 1 cosh lim 0 2 0 0 h h h h h h h h h (vi) cancelled
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2 [2] (i) , ) ( 3 x x f undefined h h h f h x h x h x f h x f x f h h h h 3 2 0 3 0 3 3 0 0 1 lim lim ) 0 ( ' lim ) ( ) ( lim ) ( ' ) ( x f is not differentiable at 0 x (ii) x x x f 2 1 ) ( 2 1 1 x x At 1 x ) 1 ( 2 ) ( lim 2 1 lim ) ( lim 2 2 lim ) ( lim 2 1 1 ) 1 ( 1 2 1 1 1 1 f x f x x f x x f f x x x x x ) ( x
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This note was uploaded on 01/09/2010 for the course MATH Math103 taught by Professor Dr.prof.mostafaamer during the Spring '09 term at German University in Cairo.

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Sheet (4) solution - Math 103 Class Work Sheet (4) Solution...

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