Question

Evaluate
$\rm \displaystyle \lim_{n\to 3}\ \frac{x^{2}-x-6}{x^{3}-3x^{2}+x-3}$

Answer 1

Solution :

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Simplification :

i ) x²- x - 6

Splitting the middle term, we get

= x² - 3x + 2x - 6

= x( x - 3 ) + 2( x - 3 )

= ( x - 3 )( x + 2 ) -----( 1 )

ii ) x³ - 3x² + x - 3

= ( x - 3 )( x² + 1 ) ----( 2 )

iii ) [x²-x-6]/(x³-3x²+x-3)

= [(x-3)(x+2)]/[(x-3)(x²+1)]

= (x+2)/(x²+1) ---( 3 )
_________________________

Now,

$\rm \displaystyle \lim_{x\to 3}\ \frac{x^{2}-x-6}{x^{3}-3x^{2}+x-3}$

= $\rm \displaystyle \lim_{x\to 3}\ \frac{x+2}{x^{2}+1}$

= (3+2)/(3²+1)

= 5/10

= 1/2

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