AP Calculus BC 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - Exam Style Questions - MCQs - New Syllabus
No-CalcQuestion
The line \( y=2 \) is a horizontal asymptote to the graph of which function?
(A) \( \displaystyle y=\frac{x+2\sin x}{x-2} \)
(B) \( \displaystyle y=\frac{2x^{2}+\sin x}{x^{2}} \)
(C) \( \displaystyle y=\frac{2x^{2}+\sin x}{4-x^{2}} \)
(D) \( \displaystyle y=\frac{2^{x}+2\sin x}{2^{x}} \)
(B) \( \displaystyle y=\frac{2x^{2}+\sin x}{x^{2}} \)
(C) \( \displaystyle y=\frac{2x^{2}+\sin x}{4-x^{2}} \)
(D) \( \displaystyle y=\frac{2^{x}+2\sin x}{2^{x}} \)
▶️ Answer/Explanation
Detailed solution
\[ \lim_{x\to\infty}\frac{x+2\sin x}{x-2}=1,\quad \lim_{x\to\infty}\frac{2x^{2}+\sin x}{x^{2}}=2,\quad \lim_{x\to\infty}\frac{2x^{2}+\sin x}{4-x^{2}}=-2,\quad \lim_{x\to\infty}\frac{2^{x}+2\sin x}{2^{x}}=1. \] Thus only option (B) has horizontal asymptote \( y=2 \). ✅ Correct: (B)
No-Calc Question
If \(f\) is a function such that \(\lim_{x\to\infty} f(x)=0\), which of the following could be an expression for \(f(x)\)?
(A) \(\dfrac{x^{2}}{x^{2}+4}\)
(B) \(\dfrac{\sin x}{x}\)
(C) \(\cos x\)
(D) \(e^{x}\)
(B) \(\dfrac{\sin x}{x}\)
(C) \(\cos x\)
(D) \(e^{x}\)
▶️ Answer/Explanation
Correct answer: (B)
\(\dfrac{\sin x}{x}\to0\) since \(|\sin x|\le1\) and \(x\to\infty\).
(A) \(\to1\); (C) oscillates; (D) \(\to\infty\).