Evaluating Limits Graphically

Show your work for each problem.

1.
The graph of $f$ is a straight line that runs through an open circle at $(3, 4)$ . There is also a filled dot at $(3, 1)$ . What is $\lim_{x \to 3} f(x)$ ?
1. a)1
2. b)3
3. c)4
4. d)does not exist
Answer: 4
Both sides of the line head for the open circle at height $4$ , so $\lim_{x \to 3} f(x) = 4$ . The filled dot at $(3, 1)$ only gives $f(3) = 1$ .
2.
The graph of $g$ is the parabola $y = x^2$ with an open circle at $(2, 4)$ and a filled dot at $(2, 0)$ . What is $\lim_{x \to 2} g(x)$ ?
Answer: 4
Near $x = 2$ the parabola heads for $2^2 = 4$ from both sides, which is the height of the open circle. So $\lim_{x \to 2} g(x) = 4$ .
3.
The graph of $f$ is the line $y = x + 2$ with an open circle at $(1, 3)$ and a filled dot at $(1, 6)$ . Find $\lim_{x \to 1} f(x)$ and compare it with $f(1)$ .
Answer: 3
1. Which height does the line approach from the left of $x = 1$ ? 3
2. Which height does the line approach from the right of $x = 1$ ? 3
3. What is $f(1)$ (the filled dot)? 6
4. So what is $\lim_{x \to 1} f(x)$ ? 3
4.
The graph of $f$ is a straight line through an open circle at $(3, 4)$ , plus a filled dot at $(3, 1)$ . What is $f(3)$ ?
1. a)4
2. b)1
3. c)3
4. d)does not exist
Answer: 1
The open circle at $(3, 4)$ is not part of the graph. The filled dot at $(3, 1)$ is, so $f(3) = 1$ , even though the limit at $3$ is $4$ .
5.
The graph of $g$ is the parabola $y = x^2$ with an open circle at $(2, 4)$ and a filled dot at $(2, 0)$ . What is $g(2)$ ?
Answer: 0
The filled dot $(2, 0)$ is the point of the graph at $x = 2$ , so $g(2) = 0$ . The open circle only shows where the parabola is heading.
6.
The graph of $f$ is the line $y = 4 - x$ , drawn without any gaps or extra dots. Find $\lim_{x \to 1} f(x)$ and check whether it equals $f(1)$ .
Answer: 3
1. Which height does the line approach from the left of $x = 1$ ? 3
2. Which height does it approach from the right of $x = 1$ ? 3
3. What is $f(1)$ ? 3
4. Is the limit equal to $f(1)$ ? yes
7.
For $x < 2$ the graph of $f$ follows $y = x$ and ends in an open circle at $(2, 2)$ . For $x \ge 2$ it follows $y = 5 - x$ , starting with a filled dot at $(2, 3)$ . What is $\lim_{x \to 2} f(x)$ ?
1. a)2
2. b)5
3. c)3
4. d)does not exist
Answer: does not exist
From the left the graph approaches $2$ , from the right it approaches $5 - 2 = 3$ . The sides disagree, so the limit does not exist, even though $f(2) = 3$ .
8.
The graph of $h$ is the line through $(0, -1)$ and $(4, 7)$ , with an open circle at $x = 2$ and no filled dot there. What is $\lim_{x \to 2} h(x)$ ?
Answer: 3
The slope is $\frac{8}{4} = 2$ and the line meets the $y$ -axis at $-1$ , so $y = 2x - 1$ . Both sides approach $2 \cdot 2 - 1 = 3$ , so $\lim_{x \to 2} h(x) = 3$ although $h(2)$ is undefined.
9.
For $x < 2$ the graph of $f$ is the line $y = x$ , ending in an open circle at $(2, 2)$ . For $x \ge 2$ it is the line $y = x + 3$ , starting with a filled dot at $(2, 5)$ . Does $\lim_{x \to 2} f(x)$ exist, and what is $f(2)$ ?
Answer: 5
1. Which height does the graph approach from the left of $x = 2$ ? 2
2. Which height does the graph approach from the right of $x = 2$ ? 5
3. Do the two sides agree, so that $\lim_{x \to 2} f(x)$ exists? no
4. What is $f(2)$ ? 5
10.
Which statement about $\lim_{x \to a} f(x)$ is always true?
1. a)If $f(a)$ is undefined, the limit at $a$ does not exist
2. b)If there is a filled dot at $x = a$ , the limit equals its height
3. c)The limit at $a$ depends only on the values of $f$ near $a$ , not at $a$
4. d)The limit at $a$ always equals $f(a)$
Answer: The limit at $a$ depends only on the values of $f$ near $a$ , not at $a$
The limit only looks at the values of $f$ near $a$ . A hole (undefined $f(a)$ ) does not stop a limit, a moved dot makes the limit differ from $f(a)$ , and at a jump there is a filled dot but no limit.
11.
Near $x = -2$ the graph of $f$ approaches an open circle at $(-2, 3)$ from both sides, and there is a filled dot at $(-2, -1)$ . Compute $\lim_{x \to -2} f(x) + f(-2)$ .
Answer: 2
$\lim_{x \to -2} f(x) = 3$ (the height the curve approaches) and $f(-2) = -1$ (the filled dot), so the sum is $3 + (-1) = 2$ .
12.
The graph of $g$ : for $x \le -1$ it is $y = x + 3$ , ending in a filled dot at $(-1, 2)$ ; for $-1 < x < 2$ it is $y = (x - 1)^2$ , starting at an open circle at $(-1, 4)$ ; for $x > 2$ it is $y = 3 - x$ . At $x = 2$ there is an open circle at $(2, 1)$ and no filled dot. Read the limits from the graph.
Answer: 1
1. What is $\lim_{x \to 0} g(x)$ ? 1
2. What is $\lim_{x \to 2} g(x)$ ? 1
3. Is $g(2)$ defined? no
4. Which height does the graph approach from the right of $x = -1$ ? 4
13.
The graph of $f(x) = \frac{x^2 - 1}{x + 1}$ looks like the line $y = x - 1$ with a hole. Use the graph to find $\lim_{x \to -1} f(x)$ .
Answer: -2
1. At which $x$ is the hole (where $f$ is undefined)? -1
2. What is the height of the line $y = x - 1$ at that $x$ ? -2
3. Is there a filled dot at $x = -1$ ? no
4. So what is $\lim_{x \to -1} f(x)$ ? -2
14.
The graph of $f$ has three special places. At $x = 1$ the left part ends at height $2$ and the right part starts at height $-1$ . At $x = 3$ the curve approaches an open circle at $(3, 4)$ from both sides, with no filled dot. At $x = 5$ the curve approaches $(5, 0)$ from both sides, but the filled dot is at $(5, 3)$ . For which $a$ does $\lim_{x \to a} f(x)$ not exist?
1. a) $a = 0$
2. b) $a = 5$
3. c) $a = 3$
4. d) $a = 1$
Answer: $a = 1$
At $x = 1$ the sides approach $2$ and $-1$ : a jump, so no limit. At $x = 3$ the limit is $4$ (a hole) and at $x = 5$ the limit is $0$ (the dot at height $3$ does not matter). At $x = 0$ nothing special happens.
15.
The graph of $f$ follows $y = 3x + k$ for $x < 2$ and $y = x^2$ for $x > 2$ , with open circles at both ends at $x = 2$ . For which value of $k$ does $\lim_{x \to 2} f(x)$ exist?
Answer: -2
The left side approaches $3 \cdot 2 + k = 6 + k$ and the right side approaches $2^2 = 4$ . The limit exists when $6 + k = 4$ , so $k = -2$ .
16.
The graph of $q$ : for $x < 2$ it is $y = x^2$ , ending in an open circle at $(2, 4)$ ; there is a filled dot at $(2, 1)$ ; for $2 < x < 5$ it is $y = 6 - x$ , between open circles at $(2, 4)$ and $(5, 1)$ ; for $x \ge 5$ it is $y = x - 2$ , starting with a filled dot at $(5, 3)$ . Find $\lim_{x \to 2} q(x)$ , $q(2)$ , and decide whether $\lim_{x \to 5} q(x)$ exists.
Answer: 4
1. What is $\lim_{x \to 2} q(x)$ ? 4
2. What is $q(2)$ ? 1
3. Which height does the graph approach from the left of $x = 5$ ? 1
4. Which height does the graph approach from the right of $x = 5$ ? 3
5. Does $\lim_{x \to 5} q(x)$ exist? no
17.
For $x < 3$ the graph of $f$ is $y = ax$ , for $x > 3$ it is $y = 12 - x$ , and there is a filled dot at $(3, 0)$ . Find the value of $a$ for which $\lim_{x \to 3} f(x)$ exists.
Answer: 3
1. Which height does the graph approach from the right of $x = 3$ ? 9
2. The left side approaches $3a$ . Which $a$ makes $3a$ equal to that height? 3
3. With this $a$ , what is $\lim_{x \to 3} f(x)$ ? 9
4. What is $f(3)$ ? 0
18.
The graph of $f$ : for $x \le 0$ it is $y = -x$ , ending in a filled dot at $(0, 0)$ ; for $0 < x < 4$ it is $y = x + 2$ , between open circles at $(0, 2)$ and $(4, 6)$ ; there is a filled dot at $(4, 1)$ ; for $x > 4$ it is $y = 10 - x$ , starting at an open circle at $(4, 6)$ . Compute $\lim_{x \to 4} f(x) + \lim_{x \to 2} f(x) + f(0)$ .
Answer: 10
1. What is $\lim_{x \to 4} f(x)$ ? 6
2. What is $\lim_{x \to 2} f(x)$ ? 4
3. What is $f(0)$ ? 0
4. Does $\lim_{x \to 0} f(x)$ exist? (It is not part of the sum.) no
5. What is the sum $\lim_{x \to 4} f(x) + \lim_{x \to 2} f(x) + f(0)$ ? 10