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$\lim_{x \rightarrow 27} \frac{x^{1/3} - 3}{x - 27}$.

Hello,

I am stuck on this one. I am sure there is a simple step but I am not seeing it. Thanks in advance for the help.

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  • $\begingroup$ Notice that $$(x^{\frac 1n})^n=x$$. $\endgroup$ – Piquito Sep 28 '16 at 0:32
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Hint:

$$(x-27)=(x^{\frac{1}{3}}-3)(x^{\frac{2}{3}}+3x^{\frac{1}{3}}+9)$$

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  • $\begingroup$ ahhh thank you so much! $\endgroup$ – Jack Sep 28 '16 at 2:11
  • $\begingroup$ how would you figure out how to factor that? just practice? $\endgroup$ – Jack Sep 28 '16 at 2:16
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Hint:

This is equivalent to

$$\lim_{x\to3}\frac{x-3}{x^3-27}$$

A simpler ratio. Factor from here.

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Hint. For any differentiable function near $a$, one has $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}=f'(a). $$ Then just apply it to $f(x)=x^{1/3}-3,\, a=27$.

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