How can we prove that $$\lim_{x\rightarrow 3} \left ( x^{3} - 3x + 2 \right ) = 20$$ using the definition with $\epsilon$ and $\delta$?
How to prove this limit using definition?
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calculus
limits
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1Let $x = 3 + \delta$, $x^3-3 x+2 = 20 + 24 \delta + 9 \delta^2 + \delta^3$. Can you complete the proof now ? – 2011-10-14
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2It is often useful to remember a fact from algebra: If you plug a number---call it "$a$"---into a polynomial and get $0$, then the polynomial is divisible by $x-a$. In this case, that means that $(x^3-3x+2)-20$ is divisible by $x-3$. I.e. it can be factored as $(x-3)(\cdots\cdots\cdots)$ (and it's not hard to figure out what goes where those dots are). You see how that's used in Chandrasekhar's answer. – 2011-10-14
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0Maybe it can be useful to prove that for continuous functions you can just plug in the value. Then you can apply this result to your polynomial. – 2011-10-14
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0@Jonas: I don't see that that helps unless you already know that this function is continuous, which is essentially what is to be proved. – 2011-10-15
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0Right, but maybe he already has a theorem that states that polynomials are continuous (using the $\epsilon$-$\delta$ definition for example, and not the limit definition). – 2011-10-15