Question

Help me solve this limit problem : Click to see the question below

Answer 1

\[\lim_{x \rightarrow \sqrt{2}} \frac{ x^9-3x^8+x^6-9x^4-4x^2-16x+84 }{x^5-3x^4-4x+12 }\]

Answer 2

i would start by doing long division

Answer 3

@dumbcow : please show me how !!

Answer 4

i would divide both numerator and denominator by x^2-2
since x- sqrt 2 is a factor, x+sqrt 2 must also be , so x^2-2 is a factor.
so, try doing long division by x^2-2 for both numerator and denominator

Answer 5

@hartnn how do you understand x- sqrt 2 is a factor ?

Answer 6

@digitalmonk , sorry i logged off before and didn't see your reply
actually long division wont help here and long division is hard to teach in this forum
anyway if you plug in sqrt2 you get -76/0 , which means sqrt2 is a vertical asymptote and limit could be infinity, -infinity , or does not exist
check right/left side limits by plugging in values on either side of sqrt2 like 1 and 2

Answer 7

when you plug in x =sqrt 2 , you get 0 in the denominator as well as numerator.
this means x-sqrt 2 is a factor.
(for example, if you put x=1 in x^2-1, you get a 0, implying x-1 is a factor)

Answer 8

@hartnn : thanks .. understood