Mathematics for Management -- Supplementary Electronic Materials

Quiz: Limits at Infinity

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1. Determine the value of the limit at (negative) infinity $\lim_{x \to -\infty} \left( 2x^2 + x - 1 \right)$ (if it exists).

$ 2 $ $ -2 $ $ -1 $ does not exist

2. Determine the value of the following limit (if it exists) $$ \lim_{x \to \infty} \frac{3}{1+x^2} \, . $$

$ 0 $ $ 1 $ $ 3 $ does not exist

3. Determine the value of the following limit (if it exists) $$ \lim_{x \to \infty} \frac{x^2 + x + 1}{(3x + 2)^2} \, . $$

$ 1 $ $ \frac{1}{3} $ $ 0 $ $ \frac{1}{9} $

4. Determine the value of the following limit (if it exists) $$ \lim_{x \to -\infty} \frac{4x^3 + 2x^2 + 3x}{-9x^2 + 5x + 5} \, . $$

does not exist $ 0 $ $ \frac{4}{9} $ $ -\frac{2}{3} $

5. Determine the value of the following limit (if it exists) $$ \lim_{x \to \infty} \frac{2x^2 + 1}{(2-x)(2+x)} \, . $$

$ -4 $ $ 2 $ $ -2 $ $ 1 $

6. What is the definition of a horizontal asymptote?

A curved line. A line that only lives on the first quadrant. A line that only touches the origin. A line that a curve approaches, as it heads towards infinity.

7. Find the equation of the horizontal asymptote (if it exists) for the function $$ f(x) \, \, = \, \, \frac{x-100}{x^2 - 100} \, . $$

does not exist $ y = 0 $ $ y = 1 $ $ y = x $

8. Find the equation of the horizontal asymptote (if it exists) for the function $$ f(x) \, \, = \, \, \frac{x^2 - 25}{x^2-x-20} \, . $$

$ y = 1 $ $ y = 0 $ $ y = \frac{5}{4} $ does not exist

9. The horizontal asymptote equals zero when

the highest exponents in the numerator and denominator are equal. the highest exponent in the numerator is greater than that in the denominator. the highest exponent in the numerator is less than that in the denominator. the denominator equals zero.

10. Suppose the total cost, $C(q)$, of producing a quantity $q$ of a product is given by the equation $C(q) = 5000 + 5q$. The average cost per unit quantity, $A(q)$, equals the total cost, $C(q)$, divided by the quantity produced, $q$. Find the limiting value of the average cost per unit as $q$ tends to $\infty$. In other words find $\lim_{q \to \infty} A(q)$.

$ 5 $ $ 5000 $ $ 6 $ does not exist

Your grade is: __ of 10

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Solution: 1d; 2a; 3d; 4a; 5c; 6d; 7b; 8a; 9c; 10a

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.