Connecting limits at infinity and horizontal asymptotes — AP Calculus BC
1. Core Definition: The Connection Between Concepts ★★☆☆☆ ⏱ 3 min
This topic connects two core Unit 1 ideas: limits at infinity (limits as $x$ grows without bound in the positive or negative direction) and horizontal asymptotes, which describe the long-run end behavior of a function’s graph. Unit 1 accounts for 10–12% of the total AP exam score, and this topic is tested in both multiple-choice and free-response sections.
Exam tip: Asymptotes only describe end behavior, not behavior at finite $x$ — functions can cross horizontal asymptotes without violating the definition.
2. Evaluating Limits at Infinity for Rational Functions ★★☆☆☆ ⏱ 4 min
A rational function is defined as $f(x) = \frac{P_n(x)}{Q_m(x)}$, where $P_n(x)$ is a polynomial of degree $n$ and $Q_m(x)$ is a polynomial of degree $m$. For large values of $|x|$, the highest-degree term dominates all lower terms, which become negligible. To evaluate the limit, divide both the numerator and denominator by the highest power of $x$ in the denominator, using the fact that $\lim_{x \to \pm \infty} \frac{1}{x^k} = 0$ for any $k>0$.
- If $n < m$ (numerator lower degree than denominator): $\lim_{x \to \pm \infty} f(x) = 0$
- If $n = m$ (degrees equal): $\lim_{x \to \pm \infty} f(x) = \frac{\text{leading coefficient of } P_n}{\text{leading coefficient of } Q_m}$
- If $n > m$ (numerator higher degree than denominator): $\lim_{x \to \pm \infty} f(x) = \pm \infty$, no finite limit
Exam tip: When evaluating limits of even-powered roots at negative infinity, remember that $\sqrt{x^2} = |x| = -x$ for $x<0$; always check the sign to avoid errors.
3. Finding All Horizontal Asymptotes ★★★☆☆ ⏱ 3 min
By definition, a function $y=f(x)$ has a horizontal asymptote at $y=L$ if either $\lim_{x \to \infty} f(x) = L$ *or* $\lim_{x \to -\infty} f(x) = L$, where $L$ is a finite number. Common misconceptions: (1) non-rational functions can have two different horizontal asymptotes, one for each end, and (2) crossing a horizontal asymptote at finite $x$ does not invalidate it.
Exam tip: Never assume a function only has one horizontal asymptote; always check both $x \to \infty$ and $x \to -\infty$ for non-rational functions.
4. Limits at Infinity for Non-Rational Functions ★★★☆☆ ⏱ 4 min
AP BC exam questions regularly ask for horizontal asymptotes of non-rational functions including products/quotients of polynomials and exponentials, logarithmic functions, and oscillating bounded functions. For indeterminate forms, use L'Hospital's Rule or the Squeeze Theorem. Key standard results to memorize:
- For $a>0$: $\lim_{x \to \infty} e^{-ax} = 0$, $\lim_{x \to \infty} e^{ax} = \infty$
- For any $p>0$: $\lim_{x \to \infty} \frac{\ln x}{x^p} = 0$ (logs grow slower than any positive power of $x$)
- If $|g(x)| \leq M$ (bounded) and $\lim_{x \to \infty} h(x) = 0$, then $\lim_{x \to \infty} g(x)h(x) = 0$ (Squeeze Theorem result)
Exam tip: For indeterminate forms at infinity ($\infty - \infty$, $0 \cdot \infty$, $\infty/\infty$), always rewrite the expression before concluding a finite limit exists.
5. AP-Style Practice: Worked Examples ★★★★☆ ⏱ 4 min
Common Pitfalls
Why: Forgets that $\sqrt{x^2} = |x| = -x$ for negative $x$, so the sign is incorrect.
Why: Believes the myth that functions cannot cross their asymptotes, which only applies to vertical asymptotes, not horizontal.
Why: Reverses the outcome of the degree rule for horizontal asymptotes.
Why: Assumes exponentials only have one horizontal asymptote and forgets to check the left end.
Why: Confuses finite number subtraction with the indeterminate form $\infty - \infty$.
Why: Forgot the Squeeze Theorem applies to products of bounded functions and functions that go to zero.