Calculus Quiz

Calculate the limit as x approaches infinity for the expression \(\frac{2x - 5\sqrt{x} - 1}{8x^2 - 1}\).

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The function f has the property that as x gets sufficiently close to 4, but not equal to 4, the values of f(x) get closer and closer to -1. Which of the following statements must be true?

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If \(f(x) = \begin{cases} e^x + e^{-x} & \text{for } 0 < x < 2 \\ x^2 - x - e^x & \text{for } 2 \leq x < 8 \end{cases}\), then calculate \(\lim_{x \to 2} f(x)\).

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If \(k \neq 0\), calculate the limit as x approaches k for the expression \(\frac{x^2 + 3x - kx - 3k}{x - k}\).

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Let f be a continuous function on the closed interval [-2, 6] such that f(-2) = 4 and f(6) = -4. Which of the following is guaranteed by the Intermediate Value Theorem?

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The graph of the function f is shown above. Which of the following statements is true?

• (A) f is continuous at x = 2

• (B) f is continuous at x = 5

• (C) f is continuous at x = 6

• (D) f is continuous at x = 8

• (E) f is continuous at x = 9

The table above gives values of continuous functions f and g and their values at selected values of x. If h is the function defined by h(x) = f(x)g(x) - 2, then lim(x→∞) h(x) = ?

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The graph of f(x) = (a - bx) / (c - x) has a horizontal asymptote of y = -6 and a vertical asymptote of x = 6. What is the value of b + c?

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The function f is given by f(x) = x^3 - 8x^2 + 20x - 12. For how many positive values of c does lim(x→c) f(x) = 4?

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lim(x→0) (x / sin x) = ?

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