$\lim_{x \to \infty} f(x) = L$, read aloud: the limit of f of x as x approaches infinity is L, means the outputs settle toward $L$ as the inputs grow without bound. On the graph that is a horizontal asymptote, the line $y = L$ the curve hugs far to the right (and $x \to -\infty$ asks the same question far to the left). The engine behind every rational case is $\dfrac{1}{x} \to 0$: a fixed number divided by something enormous is next to nothing, and so is $\dfrac{1}{x^2}$, $\dfrac{7}{x^3}$, any constant over any positive power.

For a quotient of polynomials, only the highest power on each side survives far out, so compare the degrees (the highest powers). Numerator degree LOWER than the denominator's: the limit is 0. Degrees EQUAL: divide top and bottom by that power and everything else dies, leaving the ratio of the leading coefficients (the numbers in front of the highest powers): $\lim_{x \to \infty} \dfrac{3x^2 - 5x + 1}{2x^2 + 7} = \dfrac{3}{2}$. Numerator degree HIGHER: the quotient grows without bound, $+\infty$ or $-\infty$, and there is no horizontal asymptote. The leading coefficients are compared ONLY in the equal case; comparing them when the degrees differ is the trap this lesson is built around. The constant terms never matter at all: they are the first casualties.