Limit problem

Question

Limit problem

alivejeremy · Answer

?

greatlife44 · Answer

$$\lim_{x \rightarrow \infty }\frac{ 2x^{2} }{ (x+3)^{3} }$$

greatlife44 · Answer

also, Is there a way to evaluate (x+3)^3 without having to expand it out.

anonymous · Answer

Well, the degree of the numerator is lower than the denominator. So it grows asymptotically slower as x approaches infinity. Therefore the limit will go towards 0.

greatlife44 · Answer

sorry, @Tommynaiter . so is that a general rule what's the reasoning behind that? 

it would look something like this right ? 

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greatlife44 · Answer

Was just wondering say how we can evaluate this expression say using limit laws  and thanks for your explanation.

greatlife44 · Answer

when I try to expand this out I get this 


$$\frac{ 2x^{2} }{ (x+3)^{2}(x+3) }=\frac{ 2x^{2} }{ (x^{2}+6x+9)(x+3) } $$



$$\frac{ 2x^{2} }{ x^{3}+3x^{2}+6x^{2}+18x+9x+27 }$$

anonymous · Answer

You cannot use any basic limit laws here. But it is fairly obvious that the denominator will grow exponentially faster.

anonymous · Answer

Yeap, that makes it more clear that the denominator will grow faster, therefore it will go towards 0

greatlife44 · Answer

I see, so the way I understand it now is because the numerator is much bigger than the denominator as you approach infinity the denominator approaches a very large number and when compared to the numerator is much smaller so it becomes zero