Calculus Questions with Filter

Question 5

Limits Rational Functions Medium

\( \lim\limits_{x \to 2} \frac{x^2 + x - 6}{x^2 - 4} \) is

(A) \( -\frac{1}{4} \)

(B) \( 0 \)

(C) \( 1 \)

(D) \( \frac{5}{4} \)

(E) nonexistent

Integrals Fundamental Theorem of Calculus Improper Integrals Hard

If \( \int_{1}^{x^2} f(t)dt = \frac{20x}{\sqrt{4x^2 + 21}} - 4 \), then \( \int_{1}^{\infty} f(t)dt \) is

(A) \( 6 \)

(B) \( 1 \)

(C) \( -3 \)

(D) \( -4 \)

(E) divergent

Continuity Differentiability Piecewise Functions Medium

At \( x = 3 \), the function given by \( f(x) = \begin{cases} x^2 & ,x < 3 \\ 6x - 9 & ,x \ge 3 \end{cases} \) is

(A) undefined

(B) continuous but not differentiable

(C) differentiable but not continuous

(D) neither continuous nor differentiable

(E) both continuous and differentiable

Continuity Properties Calculus Theorems Easy

If \( f \) is a continuous function on \( [a, b] \), which of the following is necessarily true?

(A) \( f' \) exists on \( (a, b) \)

(B) If \( f(x_0) \) is a maximum of \( f \), then \( f'(x_0) = 0 \)

(C) \( \lim\limits_{x \to x_0} f(x) = f\left( \lim\limits_{x \to x_0} x \right) \) for \( x_0 \in (a, b) \)

(D) \( f(x) = 0 \) for some \( x \in [a, b] \)

(E) The graph of \( f \) is a straight line