The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:

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Question: The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:

Options:

Correct Answer: Not continuous at x = 0

Solution:

The function is not continuous at x = 0 since the limit does not equal f(0).

The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:

The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:

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: Identify the point we are checking for continuity, which is x = 0.
Step 3: Find 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 but is not equal to 0.
Step 4: Calculate the limit: As x approaches 0 from the left (negative side) and from the right (positive side), f(x) = 1/x goes to negative infinity and positive infinity respectively.
Step 5: Since the limit does not exist (it goes to different values from each side), we conclude that the limit of f(x) as x approaches 0 is not equal to f(0).
Step 6: Since the limit does not equal f(0), the function is not continuous at x = 0.

Continuity – The concept of continuity at a point, which requires that the limit of the function as it approaches the point equals the function's value at that point.
Limits – Understanding how to evaluate limits, particularly at points where the function is not defined or behaves differently.