Types of Discontinuity

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Show your work for each problem.

1.
What type of discontinuity does $f(x) = \frac{x^2 - 16}{x - 4}$ have at $x = 4$ ?
1. a)no discontinuity
2. b)infinite
3. c)jump
4. d)removable
2.
$f(x) = \frac{5}{x - 7}$ has an infinite discontinuity. At which $x$ is the vertical asymptote?
3.
$f(x) = \frac{x^2 - 25}{x - 5}$ . Find the $y$ -coordinate of the hole.
4.
At $x = 3$ the left-hand limit of $f$ is $2$ and the right-hand limit is $5$ . What type of discontinuity is at $x = 3$ ?
1. a)jump
2. b)no discontinuity
3. c)infinite
4. d)removable
5.
$f(x) = \frac{x^2 - 9}{x - 3}$ has a hole at $x = 3$ . What is the $y$ -coordinate of the hole?
6.
Let $f(x) = x + 1$ for $x < 2$ and $f(x) = x + 4$ for $x \geq 2$ . Find the jump size at $x = 2$ .
7.
What type of discontinuity does $f(x) = \frac{1}{(x + 5)^2}$ have at $x = -5$ ?
1. a)jump
2. b)removable
3. c)infinite
4. d)no discontinuity
8.
Let $f(x) = 3x - 1$ for $x < 2$ and $f(x) = x^2 + 3$ for $x \geq 2$ . What is the jump size at $x = 2$ ?
9.
$f(x) = \frac{x + 2}{x - 1}$ . Find where $f$ is discontinuous and check numerically that the discontinuity is infinite.
10.
Which definition of $g(1)$ removes the discontinuity of $g(x) = \frac{x^2 + 2x - 3}{x - 1}$ at $x = 1$ ?
1. a) $g(1) = 4$
2. b) $g(1) = 1$
3. c) $g(1) = 3$
4. d) $g(1) = 0$
11.
$f(x) = \frac{x^2 + 5x + 4}{x + 1}$ is undefined at $x = -1$ . Which value should $f(-1)$ get so that $f$ is continuous?
12.
Let $g(x) = \frac{x^2 - 4x}{x - 4}$ for $x \neq 4$ and $g(4) = 1$ . Classify the discontinuity at $x = 4$ and find the value of $g(4)$ that removes it.
13.
$f(x) = \frac{x - 2}{x^2 + x - 6}$ . Find both discontinuities, classify them and find the $y$ -coordinate of the hole.
14.
Which statement correctly describes the discontinuities of $r(x) = \frac{x^2 - 4}{x^2 - 5x + 6}$ ?
1. a)removable at $x = 2$ , infinite at $x = 3$
2. b)infinite at $x = 2$ and at $x = 3$
3. c)removable at $x = 2$ and at $x = 3$
4. d)removable at $x = 3$ , infinite at $x = 2$
15.
$f(x) = \frac{x^2 - 1}{x^2 + 2x - 3}$ has a hole at $x = 1$ . What is its $y$ -coordinate? (fraction or decimal)
16.
Let $f(x) = x^2 + k$ for $x < 2$ and $f(x) = 3x - 1$ for $x \geq 2$ . Which value of $k$ removes the jump at $x = 2$ ?
17.
$f(x) = \frac{x^2 - x}{x^3 - x}$ . Find all discontinuities: the holes with their heights and the vertical asymptote.
18.
Let $f(x) = \frac{1}{x - 1}$ for $x < 3$ and $f(x) = x$ for $x \geq 3$ . Classify the discontinuity at $x = 1$ and find the jump size at $x = 3$ .