What can be said about the function f(x) = |x| at x = 0?

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Question: What can be said about the function f(x) = |x| at x = 0?

Options:

Correct Answer: Continuous

Solution:

The function f(x) = |x| is continuous at x = 0 since both left and right limits equal f(0) = 0.

What can be said about the function f(x) = |x| at x = 0?

What can be said about the function f(x) = |x| at x = 0?

Step 1: Understand the function f(x) = |x|. This means that for any value of x, f(x) gives the absolute value of x.
Step 2: Find the value of the function at x = 0. Calculate f(0) = |0| = 0.
Step 3: Determine the left limit as x approaches 0. This means looking at values of x that are slightly less than 0. For example, if x = -0.1, then f(-0.1) = |-0.1| = 0.1. As x gets closer to 0 from the left, f(x) approaches 0.
Step 4: Determine the right limit as x approaches 0. This means looking at values of x that are slightly greater than 0. For example, if x = 0.1, then f(0.1) = |0.1| = 0.1. As x gets closer to 0 from the right, f(x) also approaches 0.
Step 5: Compare the left limit and right limit at x = 0. Both limits equal 0.
Step 6: Since the left limit, right limit, and f(0) all equal 0, we conclude that the function f(x) = |x| is continuous at x = 0.

Continuity – The function is continuous at a point if the limit from the left equals the limit from the right and both equal the function's value at that point.