The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?

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Question: The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?

Options:

Correct Answer: No

Solution:

The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.

The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?

The function f(x) = { 1/x, x != 0; 0, x = 0 } is 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 if f(x) is continuous at x = 0, we need to find the limit of f(x) as x approaches 0.
Step 3: Calculate the limit: lim (x -> 0) f(x) = lim (x -> 0) (1/x). As x gets closer to 0, 1/x becomes very large (positive or negative depending on the direction).
Step 4: Since the limit does not approach a specific number (it goes to infinity), we conclude that lim (x -> 0) f(x) does not equal f(0).
Step 5: Since the limit as x approaches 0 is not equal to f(0), we say that f(x) is not continuous at x = 0.

Continuity of Functions – Understanding the definition 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 – Evaluating limits, particularly one-sided limits, to determine the behavior of a function as it approaches a specific value.