Home / AP Calculus AB 1.15 Connecting Limits at Infinity and Horizontal Asymptotes – MCQs

AP Calculus AB 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - MCQs - Exam Style Questions

No-Calc Question

If \(f\) is a function such that \(\displaystyle \lim_{x\to\infty} f(x)=0\), which of the following could be an expression for \(f(x)\)?

(A) \(\dfrac{x^{5}}{2^{x}}\)
(B) \(\dfrac{e^{x}}{x^{50}}\)
(C) \(\dfrac{x}{\ln x}\)
(D) \(\dfrac{x^{3}-4x+3}{25x^{3}+147x^{2}+98x}\)

▶️ Answer/Explanation

(A) Exponential \(2^{x}\) dominates any power \(x^{5}\) ⇒ limit \(0\).
(B) \(\dfrac{e^{x}}{x^{50}}\to\infty\).
(C) \(\dfrac{x}{\ln x}\to\infty\).
(D) Ratio of equal-degree polynomials \(\to \dfrac{1}{25}\ne 0\).

Answer: (A)

No-Calc Question

\(\ \displaystyle \lim_{x\to\infty}\frac{10-6x^{2}}{5+3e^{x}}\) is
(A) \(-2\)
(B) \(0\)
(C) \(2\)
(D) nonexistent
▶️ Answer/Explanation

As \(x\to\infty\), the denominator \(3e^{x}\) dominates any polynomial term.
Numerator behaves like \(-6x^{2}\).
Ratio \(\dfrac{-6x^{2}}{3e^{x}}\to 0\).

Answer: (B)

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