1. Sketch and analyze $f(x) = \frac{1}{x}$. - Problem: Graph $f(x) = \frac{1}{x}$ and identify discontinuities. - Formula: $f(x) = \frac{1}{x}$. - Important: The function is undefined where denominator is zero. - Work: Denominator zero at $x=0$, so discontinuity at $x=0$. - Explanation: The graph has vertical asymptote at $x=0$ and is continuous elsewhere. 2. Graph $f(x) = |x|$ and check continuity. - Problem: Is $f(x) = |x|$ continuous everywhere? - Formula: $f(x) = |x|$. - Important: Absolute value is continuous everywhere. - Work: No points of discontinuity. - Explanation: The graph is a V-shape continuous for all real $x$. 3. Sketch $f(x) = \cot(x)$ and mark discontinuities. - Problem: Identify discontinuities of $f(x) = \cot(x)$. - Formula: $\cot(x) = \frac{\cos(x)}{\sin(x)}$. - Important: Discontinuities where $\sin(x) = 0$. - Work: $\sin(x) = 0$ at $x = k\pi$, $k \in \mathbb{Z}$. - Explanation: Vertical asymptotes at multiples of $\pi$. 4. Graph $f(x) = \sqrt{x}$ and discuss continuity domain. - Problem: Domain and continuity of $f(x) = \sqrt{x}$. - Formula: $f(x) = \sqrt{x}$. - Important: Defined for $x \geq 0$. - Work: Continuous on $[0, \infty)$. - Explanation: Not defined for $x<0$, continuous on domain. 5. For $f(x) = \ln(|x|)$, show continuity. - Problem: Where is $f(x) = \ln(|x|)$ continuous? - Formula: $f(x) = \ln(|x|)$. - Important: $\ln$ defined for positive arguments. - Work: $|x| > 0$ means $x \neq 0$. - Explanation: Continuous on $(-\infty,0) \cup (0,\infty)$. 6. Sketch $f(x) = \frac{x^2 - 9}{x - 3}$ and identify discontinuity at $x=3$. - Problem: Type of discontinuity at $x=3$. - Formula: $f(x) = \frac{x^2 - 9}{x - 3} = \frac{(x-3)(x+3)}{x-3}$. - Important: Simplify for $x \neq 3$. - Work: $f(x) = x + 3$ for $x \neq 3$. - Explanation: Removable discontinuity (hole) at $x=3$. 7. Graph $f(x) = \csc x$ and discuss behavior near 0. - Problem: Behavior of $f(x) = \csc x = \frac{1}{\sin x}$ near $x=0$. - Important: $\sin x = 0$ at $x=0$. - Work: Vertical asymptote at $x=0$. - Explanation: Function tends to $\pm \infty$ near 0, discontinuous at 0. 8. Sketch piecewise function: $f(x) = \begin{cases} x + 2, & x < 0 \\ x^2, & x \geq 0 \end{cases}$ - Problem: Is $f$ continuous at 0? - Work: Left limit $= 0 + 2 = 2$. - Right limit $= 0^2 = 0$. - $f(0) = 0$. - Explanation: Limits differ, so discontinuous at 0. 9. Graph $f(x) = e^x$ and check continuity. - Problem: Is $f(x) = e^x$ continuous for all real $x$? - Important: Exponential function continuous everywhere. - Explanation: Continuous on $(-\infty, \infty)$. 10. Sketch $f(x) = -\frac{1}{x^2 + 1}$ and identify asymptotes and continuity. - Problem: Asymptotes and continuity of $f(x)$. - Important: Denominator $x^2 + 1 > 0$ for all real $x$. - Work: No vertical asymptotes. - Horizontal asymptote: $y \to 0^-$ as $x \to \pm \infty$. - Explanation: Continuous everywhere, no vertical asymptotes, horizontal asymptote at $y=0$. Final answers summarized: - $f(x) = \frac{1}{x}$ discontinuous at $x=0$. - $f(x) = |x|$ continuous everywhere. - $f(x) = \cot x$ discontinuous at $x = k\pi$. - $f(x) = \sqrt{x}$ continuous on $[0, \infty)$. - $f(x) = \ln(|x|)$ continuous on $(-\infty,0) \cup (0,\infty)$. - $f(x) = \frac{x^2 - 9}{x - 3}$ removable discontinuity at $x=3$. - $f(x) = \csc x$ discontinuous at $x = k\pi$, behavior near 0 is vertical asymptote. - Piecewise $f$ discontinuous at 0. - $f(x) = e^x$ continuous everywhere. - $f(x) = -\frac{1}{x^2 + 1}$ continuous everywhere, horizontal asymptote $y=0$.