Topic 1.15 Connecting Limits at Infinity and Horizontal Asymptotes: evaluate limits as x approaches plus or minus infinity and use them to identify horizontal asymptotes, especially for rational functions.

What this topic is asking

The College Board (Topic 1.15) wants you to evaluate limits at infinity - the end behavior of a function as $x \to +\infty$ or $x \to -\infty$ - and connect them to horizontal asymptotes. For rational functions, a quick degree comparison settles the limit, and you must handle roots and the sign of $\sqrt{x^2} = |x|$ carefully.

Limits at infinity

A key building block is $\lim_{x \to \pm\infty} \frac{1}{x^n} = 0$ for any positive power $n$: as $x$ explodes, $\frac{1}{x^n}$ vanishes.

The degree rule for rational functions

The reliable technique: divide by the highest power

Rather than memorizing only the rule, the safe method is to divide every term, top and bottom, by the highest power of $x$ in the denominator, then send the $\frac{1}{x^n}$ terms to $0$. This also handles roots if you remember $\sqrt{x^2} = |x|$, which equals $x$ for $x \to +\infty$ but $-x$ for $x \to -\infty$.

Why the degree rule works

The degree rule is not a separate fact to memorize alongside the dividing-through technique; it is the result of that technique applied in general. When you divide every term by the highest power of $x$ in the denominator, each term becomes a constant plus pieces of the form $\frac{c}{x^k}$ that vanish at infinity. If the numerator's degree is smaller, its leading term ends up divided by a higher power and goes to zero, leaving $\frac{0}{\text{constant}} = 0$. If the degrees match, the leading terms of top and bottom both survive as their coefficients, giving the ratio of leading coefficients. If the numerator's degree is larger, its leading term is divided by a smaller power and grows without bound, giving $\pm\infty$. Understanding the rule as the outcome of dividing through means you can always fall back on the reliable method when a function is not a simple polynomial ratio, such as one involving roots or different powers.

A function can have two horizontal asymptotes

The limits as $x \to +\infty$ and $x \to -\infty$ can differ, especially with roots like $\sqrt{x^2 + 1}$ where $\sqrt{x^2} = |x|$ flips sign. When the two end limits are different finite numbers, the graph has two distinct horizontal asymptotes. Always evaluate both ends if the question asks for all asymptotes. Polynomial ratios without roots usually give the same horizontal asymptote at both ends, so the two-asymptote behavior is the signature of an absolute value or an even root in the expression. A horizontal asymptote also describes only end behavior: unlike a vertical asymptote, a curve is allowed to cross its horizontal asymptote in the middle of the graph and only has to settle toward it as $x$ grows large in magnitude, a subtlety worth remembering when sketching.