1. **Problem Statement:** Find the limits of given functions as $x$ approaches specified values using tables of values and analyze right-hand and left-hand limits for parts (k), (m), and (o). 2. **Limit Definition:** The limit of $f(x)$ as $x$ approaches $a$ is $L$ if for every number $\\epsilon > 0$, there exists a number $\\delta > 0$ such that whenever $0 < |x - a| < \\delta$, then $|f(x) - L| < \\epsilon$. 3. **Right-hand limit ($\lim_{x \to a^+} f(x)$):** The value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$. 4. **Left-hand limit ($\lim_{x \to a^-} f(x)$):** The value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$. --- **Solutions:** (a) $\lim_{x \to 0} x = 0$ (b) $\lim_{x \to 1} 2x = 2(1) = 2$ (c) $\lim_{x \to 2} (-x + 1) = -(2) + 1 = -1$ (d) $\lim_{x \to -1} (1 - x) = 1 - (-1) = 2$ (e) $\lim_{x \to 0} (2x - 1) = 2(0) - 1 = -1$ (f) $\lim_{x \to 3} (x - 3) = 3 - 3 = 0$ (g) $\lim_{x \to 2} (3x^2 - 2) = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10$ (h) $\lim_{x \to 0} (5 - x - x^2) = 5 - 0 - 0 = 5$ (i) $\lim_{x \to 1} (x^3 + 1) = 1^3 + 1 = 2$ (j) $\lim_{x \to 0} (1 - x^2 - x^3) = 1 - 0 - 0 = 1$ (k) $\lim_{x \to 1} \sqrt{x^3 - 1}$ - Evaluate inside the root: $x^3 - 1$ - At $x=1$, $1^3 - 1 = 0$ - Check right-hand limit: For $x > 1$, $x^3 - 1 > 0$, so $\sqrt{x^3 - 1}$ is real and positive. - Check left-hand limit: For $x < 1$, $x^3 - 1 < 0$, so $\sqrt{x^3 - 1}$ is not real (undefined in reals). - Therefore, right-hand limit is $0$, left-hand limit does not exist. - Hence, $\lim_{x \to 1} \sqrt{x^3 - 1}$ does not exist (DNE). (m) $\lim_{x \to 1} \sqrt{x^4 - x^2 + 1}$ - Evaluate inside the root at $x=1$: $1 - 1 + 1 = 1$ - Since the expression inside the root is continuous and positive near $x=1$, limit is $\sqrt{1} = 1$ - Right-hand and left-hand limits are both $1$. (o) $\lim_{x \to 1} \frac{1 - x^2}{x + 1}$ - Factor numerator: $1 - x^2 = (1 - x)(1 + x)$ - Simplify expression: $\frac{(1 - x)(1 + x)}{x + 1} = 1 - x$ (for $x \neq -1$) - Evaluate limit: $1 - 1 = 0$ - Right-hand limit: $\lim_{x \to 1^+} (1 - x) = 0$ - Left-hand limit: $\lim_{x \to 1^-} (1 - x) = 0$ - Limit exists and equals $0$. --- **Final answers:** (k) Limit does not exist because left-hand limit is undefined. (m) Limit is $1$. (o) Limit is $0$.