1. **Problem 1: Limits from the graph of function $f(x)$** (a) Find $\lim_{x \to 1} f(x)$. - From the graph, as $x$ approaches 1 from both sides, $f(x)$ approaches just above 1. - The open circle at $(1,1)$ means $f(1)=1$ but the limit depends on nearby values. - The function values near 1 approach approximately 1 from the left and right. - So, $\lim_{x \to 1} f(x) = 1$. (b) Find $\lim_{x \to 2} f(x)$. - As $x$ approaches 2, the function approaches 2 from the left (open circle at (2,2)) and 1.5 from the right (filled dot at (2,1.5)). - Since left and right limits differ, the limit does not exist. - So, $\lim_{x \to 2} f(x)$ does not exist. (c) Find $\lim_{x \to 3} f(x)$. - Left limit at 3 is near 2 (open circle at (3,2)), right limit is 1 (filled dot at (3,1)). - Left and right limits differ, so limit does not exist. (d) Find $\lim_{x \to 4} f(x)$. - The function steps up to 1.5 at $x=4$ and rises smoothly after. - Both sides approach 1.5. - So, $\lim_{x \to 4} f(x) = 1.5$. (e) Find $\lim_{x \to +\infty} f(x)$. - The function approaches horizontal asymptote $y=3$ as $x \to +\infty$. - So, $\lim_{x \to +\infty} f(x) = 3$. (f) Find $\lim_{x \to -\infty} f(x)$. - The function starts near 0 at $x=-1$ and increases gradually. - No explicit asymptote given, assume $\lim_{x \to -\infty} f(x) = 0$. (g) Find $\lim_{x \to 3^+} f(x)$. - Right-hand limit at 3 is the filled dot at 1. - So, $\lim_{x \to 3^+} f(x) = 1$. (h) Find $\lim_{x \to 3^-} f(x)$. - Left-hand limit at 3 is near 2 (open circle). - So, $\lim_{x \to 3^-} f(x) = 2$. (i) Find $\lim_{x \to 0} f(x)$. - From graph, near $x=0$, $f(x)$ is near 0. - So, $\lim_{x \to 0} f(x) = 0$. --- 2. **Problem 2: Limits of given functions** (a) $f(x) = \sqrt{5 - x}$ - Find $\lim_{x \to 0} \sqrt{5 - x} = \sqrt{5 - 0} = \sqrt{5}$. - Find $\lim_{x \to 5^+} \sqrt{5 - x}$. - For $x > 5$, $5 - x < 0$, square root not real, so limit does not exist in reals. - Find $\lim_{x \to -5^-} \sqrt{5 - x}$. - At $x \to -5^-$, $5 - (-5) = 10$, so $\sqrt{10}$. - Find $\lim_{x \to -5} \sqrt{5 - x} = \sqrt{10}$. - Find $\lim_{x \to 5} \sqrt{5 - x}$ from left side only (since right side not real). - $\lim_{x \to 5^-} \sqrt{5 - x} = 0$. - Find $\lim_{x \to -\infty} \sqrt{5 - x}$. - As $x \to -\infty$, $5 - x \to +\infty$, so limit is $+\infty$. - Find $\lim_{x \to +\infty} \sqrt{5 - x}$. - For large $x$, $5 - x$ negative, no real values, limit does not exist. (b) $$f(x) = \begin{cases} \frac{x-5}{|x-5|}, & x \neq 5 \\ 0, & x=5 \end{cases}$$ - Find $\lim_{x \to 0} f(x)$. - For $x$ near 0, $x \neq 5$, so $f(x) = \frac{x-5}{|x-5|}$. - At $x=0$, $0-5 = -5$, $|0-5|=5$, so $f(0) = -1$. - Limit as $x \to 0$ is $-1$. - Find $\lim_{x \to 5^+} f(x)$. - For $x > 5$, $x-5 > 0$, so $f(x) = \frac{x-5}{x-5} = 1$. - Find $\lim_{x \to -5^-} f(x)$. - For $x < -5$, $x-5 < 0$, so $f(x) = -1$. - Find $\lim_{x \to -5} f(x)$. - Left and right limits at $-5$ are both $-1$, so limit is $-1$. - Find $\lim_{x \to 5} f(x)$. - Left limit at 5: $x < 5$, $f(x) = -1$. - Right limit at 5: $x > 5$, $f(x) = 1$. - Limits differ, so limit does not exist. - Find $\lim_{x \to -\infty} f(x)$. - For very large negative $x$, $x-5 < 0$, so $f(x) = -1$. - Find $\lim_{x \to +\infty} f(x)$. - For very large positive $x$, $x-5 > 0$, so $f(x) = 1$. --- **Final answers:** 1. (a) 1 (b) Does not exist (c) Does not exist (d) 1.5 (e) 3 (f) 0 (g) 1 (h) 2 (i) 0 2. (a) $\lim_{x \to 0} = \sqrt{5}$ $\lim_{x \to 5^+}$ does not exist (not real) $\lim_{x \to -5^-} = \sqrt{10}$ $\lim_{x \to -5} = \sqrt{10}$ $\lim_{x \to 5^-} = 0$ $\lim_{x \to -\infty} = +\infty$ $\lim_{x \to +\infty}$ does not exist (not real) (b) $\lim_{x \to 0} = -1$ $\lim_{x \to 5^+} = 1$ $\lim_{x \to -5^-} = -1$ $\lim_{x \to -5} = -1$ $\lim_{x \to 5}$ does not exist $\lim_{x \to -\infty} = -1$ $\lim_{x \to +\infty} = 1$