Question

Lim x-1 x³-2x²+3x-2/(2x^4-3x+1)

Answer 1

Since the limit becomes the undetermined form

`\displaystyle \lim_(x\to 1) (x^3-2x^2+3x-2)/(2x^4-3x+1) \to (0)/(0)`

it means that both polynomials have a root at
`x=1`. So, we can fact both numerator and denominator:

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

`2x^4-3x+1 = (x-1)(2x^3+2x^2+2x-1)`

So, the fraction becomes

`((x-1)(x^2-x+2))/((x-1)(2x^3+2x^2+2x-1)) = (x^2-x+2)/(2x^3+2x^2+2x-1)`

Now, as x approaches 1, you have no problems anymore:

`\displaystyle \lim_(x\to 1) (x^2-x+2)/(2x^3+2x^2+2x-1) \to (2)/(5)`