To sketch a function with the given discontinuities, follow these conditions:

1. Hole (removable discontinuity) at $x = 1$:

   • The function is defined everywhere except at $x = 1$.
   • There is a point missing (a hole) at $x = 1$.
   • The function may have a limit at $x = 1$, but it does not take that value.

2. Jump discontinuity at $x = 2$:

   • The left-hand limit ($\lim_{x \to 2^-} f(x)$) and right-hand limit ($\lim_{x \to 2^+} f(x)$) exist but are not equal.
   • This creates a sudden jump in the graph at $x = 2$.

3. Vertical asymptote at $x = 3$ (infinite discontinuity):

   • The function approaches infinity ($+\infty$ or $-\infty$) as $x$ approaches 3.
   • The graph gets arbitrarily large or small near $x = 3$.

Possible Function Definition:

A function that satisfies these conditions could be:

\begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1, x < 2 \quad \text{(hole at $$x = 1$$)} \\ 2, & x = 1 \quad \text{(hole, value not included in function)} \\ 3, & x = 2^- \quad \text{(left-hand limit at $$x = 2$$)} \\ 1, & x = 2^+ \quad \text{(right-hand limit at $$x = 2$$)} \\ \frac{1}{x - 3}, & x > 2, x \neq 3 \quad \text{(vertical asymptote at $$x = 3$$)} \end{cases}$$ ### Sketching the Graph: - **At $$x = 1$$,** draw a small open circle where the hole is. - **At $$x = 2$$,** show a jump by plotting different points for the left and right limits. - **At $$x = 3$$,** show the function approaching positive or negative infinity on either side of the vertical asymptote. Would you like a visual sketch of this function? --- ### Related Questions: 1. What is the formal definition of removable and jump discontinuities? 2. How do you find vertical asymptotes of a rational function? 3. What are the differences between discontinuities in piecewise functions and rational functions? 4. Can a function have more than one removable discontinuity? 5. How do limits help classify discontinuities? **Tip:** A removable discontinuity occurs when a function is undefined at a point, but the limit exists. If you redefine the function at that point, continuity can be restored.