jmc

The horizontal asymptote of $f$ is the horizontal line that $f$ approaches as $x\to \pm \infty$. When the leading terms of the numerator and denominator have the same degree, that line is at the value equal to the ratio of the leading coefficients, namely $y=2/1=2$. Setting this equal to $f(x)$, $$f(x)=2=\frac{2x^2-5x-7}{x^2-4x+1}.$$Clearing the denominator, $$2(x^2-4x+1)=2x^2-8x+2=2x^2-5x-7⟹3x=9⟹x=\boxed{3}.$$