Determine the continuity of f(x) = { 1/x, x != 0; 0, x = 0 } at x = 0.

Determine the continuity of f(x) = { 1/x, x != 0; 0, x = 0 } at x = 0.

Correct Answer: f(x) is not continuous at x = 0.

Step 1: Understand the function f(x). It is defined as f(x) = 1/x when x is not equal to 0, and f(0) = 0 when x is equal to 0.
Step 2: To check for continuity at x = 0, we need to find the limit of f(x) as x approaches 0.
Step 3: Calculate the limit of f(x) as x approaches 0. This means we look at what happens to f(x) when x gets very close to 0 from both sides (left and right).
Step 4: As x approaches 0 from the left (negative side), f(x) = 1/x approaches negative infinity.
Step 5: As x approaches 0 from the right (positive side), f(x) = 1/x approaches positive infinity.
Step 6: Since the left-hand limit and right-hand limit do not match (one goes to negative infinity and the other to positive infinity), the limit as x approaches 0 does not exist.
Step 7: Since the limit does not exist, f(x) is not continuous at x = 0.

Continuity – The concept of continuity at a point involves checking if the limit of the function as it approaches that point equals the function's value at that point.
Limits – Understanding how to evaluate limits, especially one-sided limits, is crucial in determining continuity.