Question

Suppose g(x)= (1/x+1 if x<1 and 2x-1if x>/1. The best description concerning the continuity of g(x) is that the function:

A. has a jump discontinuity
B. has both jump and infinite discontinuity
C. has a removable discontinuity
D. is continuos
E. has an infinite discontinuity

Answer 1

The function g(x) is continuous as it has a single definition change point at x = 1, where both sides have matching limits and the function is defined. Therefore, there is no discontinuity at x = 1.

Step-by-step explanation:

The function g(x) is defined piecewise with two different expressions: (1/x + 1) if x < 1, and (2x - 1) if x ≥1. To determine the continuity of g(x), we must check its behavior around the point where the definition changes, which is at x = 1.

If we look at the limit of (1/x + 1) as x approaches 1 from the left, the limit is 2. Now, let's consider the limit of (2x - 1) as x approaches 1 from the right, which is also 2. Since the limits from both sides are equal and the value of the function at x = 1 is 2x - 1 = 1, there is no discontinuity at x = 1.

Therefore, the best description concerning the continuity of g(x) is that the function is continuous.