# Finding a limit of a rational function

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

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.

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

## Hint:

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

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

Hint:

This is equivalent to

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

A simpler ratio. Factor from here.

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 $f(x)=x^{1/3}-3,\, a=27$.