# limit of a rational function, $\lim _{x\to 1} \frac{x^3 +x-2}{1-x}$ [closed]

$$\lim _{x\to 1} \frac{x^3 +x-2}{1-x}$$ can you please tell how to solve this ?! I tried too many things, non worked ! it is not zero for sure ! mathway gave me $-4$ (no steps)

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## closed as off-topic by AlexR, Hayden, Claude Leibovici, Mathmo123, Giuseppe NegroJul 22 '14 at 11:28

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Welcome to math.SE! This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. –  user37238 Jul 22 '14 at 10:27
What methods have you tried? –  rae306 Jul 22 '14 at 10:32
note that given $P(x)=x^3+x-2$, that $P(1)=0$, so 1 is a root of $P$, and therefore, $x-1$ is a factor. –  John Joy Jul 22 '14 at 16:29

HINT:

$$x^3+x-2=x^3-1+x-1=(x-1)(x^2+x+1)+(x-1)$$

Use $\displaystyle x\to1\implies x\ne1\iff x-1\ne0$

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hopital rule will give you

$$\lim _{x\to 1} \frac{3x^2 +1}{-1}=-4$$

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