Evaluate:-lim (x3+3x2-9x-2)/(x3-x-6)x --2

the given limit is : $\underset{x\to 2}{lim}\frac{{x}^{3}+3{x}^{2}-9x-2}{{x}^{3}-x-6}$
which is of the form 0/0.
factorise the polynomial as (x-2) is a common factor of both numerator and denominator.
$=\underset{x\to 2}{lim}\frac{\left(x-2\right)\left({x}^{2}+5x+1\right)}{\left(x-2\right)\left({x}^{2}+2x+3\right)}\phantom{\rule{0ex}{0ex}}=\frac{{2}^{2}+5*2+1}{{2}^{2}+2*2+3}\phantom{\rule{0ex}{0ex}}=\frac{15}{11}$

this can be done by L'HOSPITAL rule also:
differentiate the numerator and denominator and substitute the value of x as 2.

hope this helps you.

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