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
(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)