One-Sided Limits

Show your work for each problem.

1.
What does $x \to 3^-$ mean?
1. a) $x$ approaches $3$ through values less than $3$
2. b) $x$ approaches $3$ through values greater than $3$
3. c) $x$ is exactly $3$
4. d) $x$ takes the value $-3$
Answer: $x$ approaches $3$ through values less than $3$
$x \to 3^-$ means that $x$ approaches $3$ from the left: $2.9, 2.99, 2.999, \dots$ These values are less than $3$ , and $x$ never equals $3$ .
2.
Let $f(x) = x + 1$ for $x < 1$ and $f(x) = x + 3$ for $x \ge 1$ . Find $\lim_{x \to 1^-} f(x)$ .
Answer: 2
For $x < 1$ the function is $x + 1$ , which approaches $1 + 1 = 2$ . So $\lim_{x \to 1^-} f(x) = 2$ .
3.
Let $f(x) = x + 1$ for $x < 1$ and $f(x) = x + 3$ for $x \ge 1$ . Find $\lim_{x \to 1^+} f(x)$ .
Answer: 4
For $x > 1$ the function is $x + 3$ , which approaches $1 + 3 = 4$ . So $\lim_{x \to 1^+} f(x) = 4$ .
4.
Find $\lim_{x \to 0^+} \frac{|x|}{x}$ .
Answer: 1
For positive $x$ , $\frac{|x|}{x} = \frac{x}{x} = 1$ . The function is constantly $1$ on the right of $0$ , so the right-hand limit is $1$ .
5.
Let $f(x) = x + 1$ for $x < 1$ and $f(x) = x + 3$ for $x \ge 1$ . What is $\lim_{x \to 1} f(x)$ ?
1. a)2
2. b)4
3. c)does not exist
4. d)6
Answer: does not exist
The left-hand limit is $2$ and the right-hand limit is $4$ . Since $2 \neq 4$ , the limit does not exist. The value $f(1) = 4$ does not change this.
6.
Let $f(x) = 2x$ for $x < 2$ and $f(x) = x + 6$ for $x \ge 2$ . How big is the jump at $x = 2$ (right-hand limit minus left-hand limit)?
Answer: 4
1. Find $\lim_{x \to 2^-} f(x)$ 4
2. Find $\lim_{x \to 2^+} f(x)$ 8
3. Right-hand limit minus left-hand limit: 4
7.
Suppose $\lim_{x \to 4^-} f(x) = 5$ , $\lim_{x \to 4^+} f(x) = 5$ and $f(4) = 1$ . Which statement is true?
1. a) $\lim_{x \to 4} f(x) = 6$
2. b) $\lim_{x \to 4} f(x)$ does not exist
3. c) $\lim_{x \to 4} f(x) = 5$
4. d) $\lim_{x \to 4} f(x) = 1$
Answer: $\lim_{x \to 4} f(x) = 5$
Both one-sided limits equal $5$ , so $\lim_{x \to 4} f(x) = 5$ . The point $f(4) = 1$ lies off the curve and does not change the limit.
8.
Let $g(x) = x^2 + 1$ for $x < 3$ and $g(x) = 2x + 5$ for $x \ge 3$ . Find $\lim_{x \to 3^-} g(x)$ .
Answer: 10
For $x < 3$ , $g(x) = x^2 + 1 \to 3^2 + 1 = 10$ . (From the right $2x + 5 \to 11$ , so the two-sided limit does not exist.)
9.
Let $h(x) = x^2 - 1$ for $x < 3$ , $h(3) = 0$ and $h(x) = 2x + 2$ for $x > 3$ . Find $\lim_{x \to 3} h(x)$ .
Answer: 8
1. Find $\lim_{x \to 3^-} h(x)$ 8
2. Find $\lim_{x \to 3^+} h(x)$ 8
3. Both sides agree. What is $\lim_{x \to 3} h(x)$ ? 8
10.
Does $\lim_{x \to 5} \frac{|x - 5|}{x - 5}$ exist?
Answer: The limit does not exist: from the left $-1$ , from the right $1$ .
1. Find $\lim_{x \to 5^+} \frac{|x - 5|}{x - 5}$ 1
2. Find $\lim_{x \to 5^-} \frac{|x - 5|}{x - 5}$ -1
3. Does the two-sided limit exist? no
11.
For which function are the left-hand and right-hand limits at $x = 0$ equal?
1. a) $f(x) = \frac{|x|}{x}$
2. b) $f(x) = \frac{x}{|x|} + 1$
3. c) $f(x) = \frac{x^2}{|x|}$
4. d) $f(x) = \frac{2|x|}{x}$
Answer: $f(x) = \frac{x^2}{|x|}$
$\frac{x^2}{|x|} = |x|$ , which approaches $0$ from both sides. The others jump: $\frac{|x|}{x}$ gives $-1$ and $1$ , $\frac{x}{|x|} + 1$ gives $0$ and $2$ , $\frac{2|x|}{x}$ gives $-2$ and $2$ .
12.
Let $p(x) = 4 - x$ for $x \le 2$ and $p(x) = x^2 + 1$ for $x > 2$ . How big is the jump at $x = 2$ (right-hand limit minus left-hand limit)?
Answer: 3
1. Find $\lim_{x \to 2^-} p(x)$ 2
2. Find $\lim_{x \to 2^+} p(x)$ 5
3. What is $p(2)$ ? 2
4. Right-hand limit minus left-hand limit: 3
13.
Let $f(x) = \frac{x^2 - 1}{x - 1}$ for $x < 1$ and $f(x) = 3x - 1$ for $x \ge 1$ . Find $\lim_{x \to 1} f(x)$ .
Answer: 2
1. For $x < 1$ : $\frac{x^2 - 1}{x - 1} = x + \square$ . What goes in the box? 1
2. Find $\lim_{x \to 1^-} f(x)$ 2
3. Find $\lim_{x \to 1^+} f(x)$ 2
4. What is $\lim_{x \to 1} f(x)$ ? 2
14.
Let $f(x) = kx + 1$ for $x < 2$ and $f(x) = x^2 + 3$ for $x \ge 2$ . Find $k$ so that $\lim_{x \to 2} f(x)$ exists.
Answer: 3
1. Find $\lim_{x \to 2^+} f(x)$ 7
2. The left-hand limit is $2k + 1$ . From $2k + 1 = 7$ , what is $2k$ ? 6
3. What is $k$ ? 3
15.
Does $\lim_{x \to 3} \frac{x^2 - 9}{|x - 3|}$ exist?
Answer: The limit does not exist: from the left $-6$ , from the right $6$ .
1. Find $\lim_{x \to 3^+} \frac{x^2 - 9}{|x - 3|}$ 6
2. Find $\lim_{x \to 3^-} \frac{x^2 - 9}{|x - 3|}$ -6
3. Does the two-sided limit exist? no
16.
Let $f(x) = cx$ for $x < 2$ and $f(x) = x^2 + c$ for $x \ge 2$ . For which value of $c$ does $\lim_{x \to 2} f(x)$ exist?
1. a)2
2. b)8
3. c)-4
4. d)4
Answer: 4
Left-hand limit: $2c$ . Right-hand limit: $2^2 + c = 4 + c$ . From $2c = 4 + c$ we get $c = 4$ . Check: left $8$ , right $8$ .
17.
Find $\lim_{x \to 4^-} \frac{x^2 - 16}{|x - 4|}$ .
Answer: -8
$x^2 - 16 = (x - 4)(x + 4)$ . For $x < 4$ , $\frac{(x - 4)(x + 4)}{-(x - 4)} = -(x + 4) \to -8$ .
18.
Let $f(x) = x^2$ for $x < 1$ , $f(x) = 2x - 1$ for $1 \le x < 3$ and $f(x) = 10 - x$ for $x \ge 3$ . Find $\lim_{x \to 1} f(x)$ , then the jump at $x = 3$ (right-hand limit minus left-hand limit).
Answer: 2
1. Find $\lim_{x \to 1^-} f(x)$ 1
2. Find $\lim_{x \to 1^+} f(x)$ 1
3. Both sides agree. What is $\lim_{x \to 1} f(x)$ ? 1
4. Find $\lim_{x \to 3^-} f(x)$ 5
5. Find $\lim_{x \to 3^+} f(x)$ 7
6. Right-hand limit minus left-hand limit at $x = 3$ : 2