Mathematics for Management -- Supplementary Electronic Materials

Quiz: One-Sided Limits

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1. Determine the value of the following one-sided limit or indicate in which sense the limit tends to infinity (infinite limit): $$ \lim_{x \to 0^-} \frac{1}{x} \, . $$

$ \infty $ $ 0 $ $ -\infty $ $ 1 $

2. Determine the value of the following one-sided limit or indicate in which sense the limit tends to infinity (infinite limit): $$ \lim_{x \to \frac{1}{5}^+} \frac{x}{10x - 2} \, . $$

$ \frac{1}{2} $ $ \frac{1}{10} $ $ \infty $ $ -\infty $

3. Determine the value of the following one-sided limit or indicate in which sense the limit tends to infinity (infinite limit): $$ \lim_{x \to -\frac{1}{2}^{-}} \frac{2x^2 - 3x - 2}{2x + 1} \, . $$

$ \infty $ $ -\infty $ $ 1 $ $ -\frac{5}{2} $

4. Determine the value of the following one-sided limit or indicate in which sense the limit tends to infinity (infinite limit): $$ \lim_{x \to 5^-} \frac{|x-5|}{x-5} \, . $$

$ 1 $ $ \infty $ $ -\infty $ $ -1 $

5. etermine the value of the one-sided limit $\lim_{x \to 3^+} f(x)$ or indicate in which sense the limit tends to infinity (infinite limit) for the function $f(x)$ given by $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} -5x+7 \, , & & \text{if $x < 3$} \\ x^2 - 16 \, , & & \text{if $x \geq 3$} \end{array} \right. $$

$ 6 $ $ -6 $ $ -7 $ $ -8 $

6. Determine the value of the one-sided limit $\lim_{x \to 2^+} f(x)$ or indicate in which sense the limit tends to infinity (infinite limit) for the function $f(x)$ given by $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} -x+10 \, , & & \text{if $|x| \leq 2$} \\ 2x^2 \, , & & \text{if $|x| > 2$} \end{array} \right. $$

$ 1 $ $ 8 $ $ 10 $ $ 0 $

7. In order for a line $x = a$ to be a vertical asymptote of $f(x)$, which of the following must be true?

$ \lim_{x \to a^+} f(x) = \infty $ $ \lim_{x \to \infty} f(x) = a $ $ \lim_{x \to \infty} f(x) = \infty $ $ \lim_{x \to a^+} f(x) = a $

8. Determine the equation of the vertical asymptote for the function $f(x)$ given by $$ f(x) \, \, = \, \, \frac{x-2}{x+5} \, . $$

$ x = -\frac{2}{5} $ $ x = -5 $ $ x = -\frac{5}{2} $ $ x = 2 $

9. How many vertical asymptotes does the following function have? $$ f(x) \, \, = \, \, \frac{(x+3)(x+2)}{(x-2)(x+1)} \, . $$

none $ 1 $ $ 2 $ $ 3 $

10. Which of the following represents the vertical asymptotes for the function $$ f(x) \, \, = \, \, \frac{5}{x^4 - 2x^3 - 8x^2} $$

$ x = -4, 0, 2 $ $ x = -2, 0, 4 $ $ x = -4, 2 $ $ x = -2, 4 $

Your grade is: __ of 10

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

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