Answer
$=\dfrac {2}{3}$
Work Step by Step
Factor each polynomial:
$\lim _{x\rightarrow 1}\dfrac {x^{3}-x^{2}+x-1}{x^{4}-x^{3}+2x-2}
\\=\lim _{x\rightarrow 1}\dfrac {x^{2}\left( x-1\right) +\left( x-1\right) }{x^{3}\left( x-1\right) +2\left( x-1\right) }
\\=\lim _{x\rightarrow 1}\dfrac {\left( x-1\right) \left( x^{2}+1\right) }{\left( x-1\right) \left( x^{3}+2\right) }$
Cancel the common factors:
$=\lim _{x\rightarrow 1}\dfrac {\left( x^{2}+1\right) }{\left( x^{3}+2\right) }
\\=\dfrac {1+1}{1+2}
\\=\dfrac {2}{3}$
