AP Calculus BC 1.8 Determining Limits Using the Squeeze Theorem - MCQs - Exam Style Questions
No-Calc Question
Let \(f\) and \(g\) be the functions defined by \[ f(x)=\frac{x^2-25}{x^2-9}, \qquad g(x)=\frac{x+5}{2x+4}. \] If the function \(h\) satisfies \(f(x)\le h(x)\le g(x)\) for \(1\le x\le 5\), what is \(\displaystyle \lim_{x\to 3} h(x)\)?
(A) \(\tfrac{1}{2}\)
(B) \(\tfrac{4}{5}\)
(C) \(1\)
(D) The limit cannot be determined from the given information.
(B) \(\tfrac{4}{5}\)
(C) \(1\)
(D) The limit cannot be determined from the given information.
▶️ Answer/Explanation
Detailed solution
Compute the limit of the upper bound: \[ \lim_{x\to 3} g(x)=\frac{3+5}{2\cdot 3+4}=\frac{8}{10}=\frac{4}{5}. \]
Near \(x=3\), \(f(x)=\dfrac{(x-5)(x+5)}{(x-3)(x+3)}\) is unbounded, but \(h\) is squeezed between \(f\) and \(g\) on \([1,5]\).
Given the squeeze and the finite bound approaching \(\tfrac{4}{5}\), the only consistent finite limit for \(h\) is \(\tfrac{4}{5}\).
✅ Answer: (B)