Answer
$\dfrac {a}{b}$
Work Step by Step
$\lim _{x\rightarrow 1}\dfrac {x^{a}-1}{x^{b}-1}=\dfrac {\dfrac {d}{dx}\left( x^{a}-1\right) }{\dfrac {d}{dx}\left( x^{b}-1\right) }=\dfrac {a\times x^{a-1}}{b\times x^{b-1}}=\dfrac {a\times 1}{b\times 1}=\dfrac {a}{b}$