Determine the continuity of the function f(x) = |x| at x = 0. (2020)

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Question: Determine the continuity of the function f(x) = |x| at x = 0. (2020)

Options:

Correct Answer: Continuous

Exam Year: 2020

Solution:

The function f(x) = |x| is continuous at x = 0 since both the left-hand limit and right-hand limit equal f(0) = 0.

Determine the continuity of the function f(x) = |x| at x = 0. (2020)

Determine the continuity of the function f(x) = |x| at x = 0. (2020)

Step 1: Understand what continuity means. A function is continuous at a point if the limit as you approach that point from both sides equals the function's value at that point.
Step 2: Identify the function we are working with, which is f(x) = |x|.
Step 3: Find the value of the function at x = 0. Calculate f(0) = |0| = 0.
Step 4: Calculate the left-hand limit as x approaches 0. This means finding the limit of f(x) as x gets closer to 0 from the left (negative side). Since f(x) = |x|, as x approaches 0 from the left, f(x) approaches 0.
Step 5: Calculate the right-hand limit as x approaches 0. This means finding the limit of f(x) as x gets closer to 0 from the right (positive side). Again, as x approaches 0 from the right, f(x) approaches 0.
Step 6: Compare the left-hand limit, right-hand limit, and the function value at x = 0. Both limits equal 0, and f(0) also equals 0.
Step 7: Conclude that since the left-hand limit, right-hand limit, and f(0) are all equal, the function f(x) = |x| is continuous at x = 0.

Continuity of Functions – Understanding the definition of continuity at a point, which requires that the limit from the left and right at that point equals the function's value at that point.
Absolute Value Function – Recognizing the properties of the absolute value function, particularly its behavior around zero.