The normal to y = f ( x ) at x = 0 is 3 x − y + 3 = 0 . Find lim x → 1 ( x − 1 ) 2 [ f ( ( x − 1 ) 2 ) − 5 f ( 4 ( x − 1 ) 2 ) + 4 f ( 7 ( x − 1 ) 2 ) ] − 1 .

Question

The normal to y = f ( x ) at x = 0 is 3 x − y + 3 = 0 . Find lim x → 1 ​ ( x − 1 ) 2 [ f ( ( x − 1 ) 2 ) − 5 f ( 4 ( x − 1 ) 2 ) + 4 f ( 7 ( x − 1 ) 2 ) ] − 1 .

TestHub · Accepted Answer

The normal slope is 3, so f ′ ( 0 ) = − 1/3 . Put u = ( x − 1 ) 2 . The bracket equals ( 1 − 5 + 4 ) f ( 0 ) + ( 1 − 20 + 28 ) f ′ ( 0 ) u + o ( u ) = 9 f ′ ( 0 ) u + o ( u ) . The limit is 1/ ( 9 f ′ ( 0 )) = − 1/3 .

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