The function f(x) = e^1/x - 1/e^1/x + 1, x ≠ 0, 0, x = 0

• (a) is continuous at x = 0
• (b) is not continuous at x = 0
• (c) is not continuous at x = 0 but can be made continuous at x = 0
• (d) none of these

The function given is f(x) = e^(1/x) - 1/e^(1/x) + 1 for x ≠ 0 and f(0) = 0. To determine its continuity at x = 0, we need to check if the limit of f(x) as x approaches 0 equals f(0).

Analyzing the Limit

As x approaches 0, the term e^(1/x) grows very large (approaches infinity) when x is positive and approaches 0 when x is negative. Therefore:

• For x > 0: f(x) approaches infinity.
• For x < 0: f(x) approaches 0.

Conclusion on Continuity

Since the left-hand limit (0) and the right-hand limit (infinity) do not match, the limit of f(x) as x approaches 0 does not equal f(0). Thus, the function is not continuous at x = 0.

Final Answer

The correct choice is (b) is not continuous at x = 0.