Question: (a) Which statement below best corresponds to the meaning of the following?
$$\lim_{x \to x_0} f(x) = f(x_0)$$
○ f is continuous at $x_0$.
○ $x_0$ is in the domain of $f$, and $f$ has a jump discontinuity at $x_0$.
○ $x_0$ is in the domain of $f$, and $f$ has a removable discontinuity at $x_0$.
○ $x_0$ is not in the domain of $f$.
For which values of $x_0$ does this statement appear to be true? Select all that apply.
□ $x_0 = -2$
□ $x_0 = 2$
□ $x_0 = -1$
□ $x_0 = 3$
□ $x_0 = 0$
□ $x_0 = 4$
□ $x_0 = 1$
□ None
(b) Which statement below best corresponds to the meaning of the following?
$$\lim_{x \to x_0} f(x) = f(x_0)$$ and both
$$\lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0}$$ and $$\lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0}$$ exist, but
$$\lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} \neq \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0}$$
○ $f$ is continuous at $x_0$, but $f$ does not have a tangent line at $x_0$.
○ $f$ is continuous at $x_0$, but $f$ has a vertical tangent line at $x_0$.
○ $f$ is continuous at $x_0$, and $f$ has a non-vertical tangent line at $x_0$.
○ $f$ is not continuous at $x_0$, and so $f$ does not have a tangent line at $x_0$.
For which values of $x_0$ does this statement appear to be true? Select all that apply.
$x_0 = -2$
$x_0 = -1$
$x_0 = 0$
$x_0 = 1$
$x_0 = 2$
$x_0 = 3$
$x_0 = 4$
None
(c) Which statement below best corresponds to the meaning of the following?
$$\lim_{x \to x_0} f(x) = f(x_0)$$ and either
$$\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = \infty$$ or
$$\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = -\infty$$
○ $f$ is continuous at $x_0$, but $f$ does not have a tangent line at $x_0$.
○ $f$ is continuous at $x_0$, but $f$ has a vertical tangent line at $x_0$.
○ $f$ is continuous at $x_0$, and $f$ has a non-vertical tangent line at $x_0$.
○ $f$ is not continuous at $x_0$, and so $f$ does not have a tangent line at $x_0$.
For which values of $x_0$ does this statement appear to be true? Select all that apply.
☐ $x_0 = -2$
☐ $x_0 = 2$
☐ $x_0 = -1$
☐ $x_0 = 3$
☐ $x_0 = 0$
☐ $x_0 = 4$
☐ $x_0 = 1$
☐ None
(d) From the choices of $x_0$ below, choose all at which $f$ appears to be differentiable.
$x_0 = -2$
$x_0 = 2$
$x_0 = -1$
$x_0 = 3$
$x_0 = 0$
$x_0 = 4$
$x_0 = 1$
None
1. **Problem Statement:**
We analyze the meaning of the limit and continuity statements given for a function $f$ at points $x_0$.
2. **Part (a):**
The statement $$\lim_{x \to x_0} f(x) = f(x_0)$$ means that the limit of $f(x)$ as $x$ approaches $x_0$ equals the function value at $x_0$.
- This is the definition of continuity at $x_0$.
- Therefore, the correct statement is: **$f$ is continuous at $x_0$.**
3. **Part (a) values:**
- The values of $x_0$ where this holds true are those where $f$ is continuous.
- Without the explicit function or graph, we cannot determine exact values, but the question implies selecting all $x_0$ where $f$ is continuous.
4. **Part (b):**
The statement:
$$\lim_{x \to x_0} f(x) = f(x_0)$$ and both
$$\lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0}$$ and $$\lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0}$$ exist but are not equal.
- This means $f$ is continuous at $x_0$ but the left and right derivatives differ.
- Hence, $f$ is continuous but does not have a tangent line at $x_0$ (because the derivative is not unique).
5. **Part (b) values:**
- Select all $x_0$ where $f$ is continuous but has a corner or cusp (non-differentiable point with different one-sided derivatives).
6. **Part (c):**
The statement:
$$\lim_{x \to x_0} f(x) = f(x_0)$$ and either
$$\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = \infty$$ or $$-\infty$$.
- This means $f$ is continuous at $x_0$ but the derivative tends to infinity or negative infinity.
- This corresponds to a vertical tangent line at $x_0$.
7. **Part (c) values:**
- Select all $x_0$ where $f$ is continuous and has a vertical tangent.
8. **Part (d):**
Choose all $x_0$ where $f$ appears differentiable.
- Differentiability requires continuity and equal left and right derivatives (finite).
**Summary:**
- (a) corresponds to continuity.
- (b) corresponds to continuity with a corner (no tangent line).
- (c) corresponds to continuity with vertical tangent.
- (d) corresponds to differentiability.
Without the explicit function or graph, exact $x_0$ values cannot be determined here.
**Final answers depend on the graph or function data provided.**