Divide through all terms by the highest power of [tex]x[/tex]:
[tex]\dfrac{12x^8-3}{3x^8-2x^7}=\dfrac{12-\frac3{x^8}}{3-\frac2x}[/tex]
As [tex]x\to\infty[/tex], the terms with [tex]x[/tex] in the denominator will vanish, leaving us with
[tex]\displaystyle\lim_{x\to\infty}\frac{12x^8-3}{3x^8-2x^7}=\frac{12}3=4[/tex]
