A removable discontinuity occurs at $$x=c$$ when $${\lim_{x \to c} f(x)}$$ exists but is not equal to $$f(c)$$. Here, the limit as $$x$$ approaches 1 exists and is 4, but the function value $$f(1)$$ is 6. This perfectly matches the definition of a removable discontinuity, which would appear as a 'hole' in the graph at $$(1, 4)$$, with a point at $$(1, 6)$$.

A is incorrect because for continuity, the limit must equal the function value. B is incorrect because a jump discontinuity requires the left and right-sided limits to be different. D is incorrect because an infinite discontinuity requires the limit to be infinite.

The given $$x$$ values ($$-3.1, -3.01, -3.001$$) are all less than -3 and are approaching -3. This constitutes an approach from the left. The corresponding $$f(x)$$ values are approaching 5. Therefore, the data suggests that the left-sided limit is 5, which is written as $${\lim_{x \to -3^-} f(x) = 5}$$.

A is incorrect as the data is for a left-sided approach. C incorrectly swaps the roles of $$x$$ and $$f(x)$$. D is a two-sided limit, and we only have evidence for the left side.

When the one-sided limit of a function as $$x$$ approaches a finite number $$c$$ is either positive or negative infinity, it signifies the presence of a vertical asymptote at $$x=c$$. In this case, as $$x$$ approaches 1, the function's values are unbounded, indicating a vertical asymptote at $$x=1$$. The fact that the limits are different from each side describes the behavior on either side of the asymptote.

A describes end behavior, not behavior at a finite point. B describes a discontinuity where both one-sided limits are finite but different. D is incorrect because the function is not continuous where it has an infinite limit.

To find the left-sided limit ($$x \to 2^-$$), we use the piece of the function defined for $$x < 2$$, which is $$f(x) = x^2$$. So, $${\lim_{x \to 2^-} x^2 = 2^2 = 4}$$. To find the right-sided limit ($$x \to 2^+$$), we use the piece of the function defined for $$x > 2$$, which is $$f(x) = 3x - 2$$. So, $${\lim_{x \to 2^+} (3x-2) = 3(2) - 2 = 4}$$. Both one-sided limits are equal to 4.

The other choices contain incorrect calculations for one or both of the one-sided limits.

The left-sided limit (-1) and the right-sided limit (1) are not equal, so the function has a jump discontinuity at $$x=0$$. The definition of continuity from the right at a point $$c$$ is $${\lim_{x \to c^+} f(x) = f(c)}$$. In this case, $${\lim_{x \to 0^+} f(x) = 1}$$ and $$f(0) = 1$$, so the function is continuous from the right. The function is not continuous from the left because $${\lim_{x \to 0^-} f(x) = -1 \neq f(0)}$$. Thus, B is the most complete and accurate description.

A is false due to the jump. C is false because the two-sided limit doesn't exist. D is false because it is not continuous from the left.

The phrase 'as $$x$$ becomes large in the negative direction' means $$x$$ is approaching negative infinity ($$x \to -\infty$$). The phrase 'its values approach 4' means the function's output approaches 4. Therefore, the correct notation is $${\lim_{x \to -\infty} g(x) = 4}$$.

A and C describe behavior near the finite value $$x=4$$. D describes behavior as $$x$$ approaches positive infinity and incorrectly states the limit value is -4.

The table of values shows that as $$x$$ approaches 1 from the left (values like 0.9, 0.99, 0.999), the values of $$f(x)$$ get closer to 5. As $$x$$ approaches 1 from the right (values like 1.1, 1.01, 1.001), the values of $$f(x)$$ also get closer to 5. Since the function approaches the same value from both sides, the limit is 5.

B is incorrect because the values are approaching a finite number. C incorrectly reverses the roles of the input and the limit value. D is incorrect because the existence of a limit at a point does not depend on the function's value at that point.

The verbal description indicates that as the input $$x$$ approaches -1, the output $$g(x)$$ approaches 7. This is the definition of a two-sided limit. The notation $${\lim_{x \to -1} g(x) = 7}$$ correctly represents this statement.

A represents the function value at a point, not the limit. B reverses the roles of the input and the limit value. D describes a situation where the right-sided limit is 7 but the left-sided limit is not, so the two-sided limit would not exist.

The statement $${\lim_{x \to 4} f(x) = 2}$$ describes the behavior of the function $$f(x)$$ as $$x$$ gets close to 4. It means the function's values approach 2. The value of $$f(4)$$ itself is not relevant to the limit. Choice B correctly states this concept.

A is incorrect because the limit of a function as $$x$$ approaches a point does not depend on the value of the function at that point. C incorrectly swaps the roles of $$x$$ and $$f(x)$$. D is too strong; the values only need to approach 2, not be exactly 2.

For a two-sided limit to exist, the left-sided limit and the right-sided limit must exist and be equal. Since both the left-sided and right-sided limits as $$x$$ approaches 2 are equal to 5, the two-sided limit $${\lim_{x \to 2} f(x)}$$ must exist and be equal to 5.

A is not necessarily true; the value of the function at the point is independent of the limit. B is not necessarily true; for continuity, we would also need to know that $$f(2) = 5$$. D is incorrect; asymptotes involve infinite limits, not finite ones.