Question

To make the function g(x) continuous at x=2, we need to ensure that the left-hand limit as x approaches 2 from the left (x<2) is equal to the right-hand limit as x approaches 2 from the right (x>2) and that both limits are equal to g(2).

Answer 1

To ensure that g(x) is continuous at x=2, both the left-hand and right-hand limits as x approaches 2 must equal g(2). The function f(x) as a horizontal line over a specific range illustrates continuity within that domain. The concept of an asymptote in functions like y = 1/x demonstrates discontinuity.

Step-by-step explanation:

To make the function g(x) continuous at x=2, you are entirely correct that we need to establish that the left-hand limit as x approaches 2 from the left matches the right-hand limit as x approaches 2 from the right, and that both of these limits equal g(2). This condition ensures the absence of a discontinuity at the point x=2.

A graph of f(x) that is a horizontal line within a restricted domain shows us an example of a function that is continuous within that domain, as there are no breaks or jumps in its values. If such a function represents a continuous probability density function, the area under the curve between two points represents the probability of a variable falling within that range, wherein PROBABILITY = AREA.

As for the function y = 1/x, which has an asymptote at x=0, this shows that y cannot be zero because it would imply that x tends towards infinity. Therefore, as x approaches 0, y will not be defined, indicating a discontinuity at that point.