Can the graph of a rational function coincide with its slant asymptote?

Asked by Varunasav | 28th Sep, 2014, 11:25: AM

Expert Answer:

C o n s i d e r space t h e space r a t i o n a l space f u n c t i o n comma y equals fraction numerator x squared minus 2 x plus 1 over denominator x plus 2 end fraction T h e space g r a p h space o f space t h e space a b o v e space f u n c t i o n space i s space a s space f o l l o w s :
 
A t space x equals minus 2 comma space t h e r e space i s space a space v e r t i c a l space a s y m p t o t e.
 
 
A s space x space i n c r e a s e s comma space t h e space s a m e space g r a p h space a p p e a r s space a s space f o l l o w s :
 
F r o m space t h e space g r a p h space w e space c a n space s e e space t h a t space t h e r e space i s space a space s h a r p space b e n d space i n space t h e space c u r v e space n e a r space y minus a x i s. B y space d i v s i o n space a l g o r i t h m comma space w e space h a v e comma fraction numerator x squared minus 2 x plus 1 over denominator x plus 2 end fraction equals x minus 4 plus fraction numerator 9 over denominator x plus 2 end fraction N o w comma space f o r space l a r g e r space v a l u e s space o f space x comma space w e space h a v e comma limit as x rightwards arrow infinity of fraction numerator x squared minus 2 x plus 1 over denominator x plus 2 end fraction equals limit as x rightwards arrow infinity of x minus 4 plus limit as x rightwards arrow infinity of fraction numerator 9 over denominator x plus 2 end fraction A s comma space limit as x rightwards arrow infinity of fraction numerator 9 over denominator x plus 2 end fraction equals 0 comma space w e space h a v e comma fraction numerator x squared minus 2 x plus 1 over denominator x plus 2 end fraction equals x minus 4 T h u s comma space t h e space l i n e a r space e q u a t i o n space y equals x minus 4 space i s space a n space o b l i q u e space a s y m p t o t e. T h u s comma space a t space t h e space v e r t i c a l space a s y m p t o t e comma space x equals minus 2 comma space t h e space g r a p h space o f space t h e space o r i g i n a l space f u n c t i o n fraction numerator x squared minus 2 x plus 1 over denominator x plus 2 end fraction space l o o k s space l i k e space t h e space p r o p e r space f u n c t i o n space fraction numerator 9 over denominator x plus 2 end fraction O b s e r v e space t h e space f o l l o w i n g space g r a p h :

Answered by Vimala Ramamurthy | 29th Sep, 2014, 11:33: AM

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