Question: Limits and continuity: Unit test
26 of 30 questions
Question number 26
The function
$h$
is defined for all real numbers.
Function $h$ is graphed. The x-axis goes from negative 8 to 8. The graph consists of a curve and a closed circle. The curve starts in quadrant 2, moves downward to $(-1.5, 1.5)$, moves upward through an open circle at $(0, 2)$ to about $(4.5, 3.8)$, moves downward, and ends at $(8, 0)$. The closed circle is plotted at $(0, -7)$.
What is a reasonable estimate for
$$\displaystyle\lim_{x\to 0}h(x)$$?
Choose 1 answer:
A
$0$
B
$-7$
C
$2$
D
The limit doesn't exist
1. **State the problem:** We need to find the limit $$\lim_{x\to 0} h(x)$$ based on the graph description of the function $h$.
2. **Recall the definition of a limit:** The limit $$\lim_{x\to a} f(x)$$ is the value that $f(x)$ approaches as $x$ approaches $a$ from both sides.
3. **Analyze the graph near $x=0$:**
- The curve has an open circle at $(0, 2)$, meaning $h(0)$ is not equal to 2.
- There is a closed circle at $(0, -7)$, meaning $h(0) = -7$.
- The limit depends on the values of $h(x)$ as $x$ approaches 0, not the value at 0.
4. **Look at the behavior of $h(x)$ as $x$ approaches 0:**
- From the left and right, the curve approaches the open circle at $(0, 2)$.
- This means $$\lim_{x\to 0} h(x) = 2$$.
5. **Conclusion:** The limit exists and equals 2, even though $h(0) = -7$.
**Final answer:** $$\boxed{2}$$