One-Sided Limits

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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 $-3$
2.
Let $f(x) = x + 1$ for $x < 1$ and $f(x) = x + 3$ for $x \ge 1$ . Find $\lim_{x \to 1^-} f(x)$ .
3.
Let $f(x) = x + 1$ for $x < 1$ and $f(x) = x + 3$ for $x \ge 1$ . Find $\lim_{x \to 1^+} f(x)$ .
4.
Find $\lim_{x \to 0^+} \frac{|x|}{x}$ .
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
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)?
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) $\lim_{x \to 4} f(x) = 1$
8.
Let $g(x) = x^2 + 1$ for $x < 3$ and $g(x) = 2x + 5$ for $x \ge 3$ . Find $\lim_{x \to 3^-} g(x)$ .
9.
Let $h(x) = x^2 - 1$ for $x < 3$ , $h(3) = 0$ and $h(x) = 2x + 2$ for $x > 3$ . Find $\lim_{x \to 3} h(x)$ .
10.
Does $\lim_{x \to 5} \frac{|x - 5|}{x - 5}$ exist?
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) $f(x) = \frac{2|x|}{x}$
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)?
13.
Let $f(x) = \frac{x^2 - 1}{x - 1}$ for $x < 1$ and $f(x) = 3x - 1$ for $x \ge 1$ . Find $\lim_{x \to 1} f(x)$ .
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.
15.
Does $\lim_{x \to 3} \frac{x^2 - 9}{|x - 3|}$ exist?
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
17.
Find $\lim_{x \to 4^-} \frac{x^2 - 16}{|x - 4|}$ .
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).