Is the function f(x) = |x|/x continuous at x = 0?
Correct Answer: f(x) is not continuous at x = 0.
- Step 1: Understand the function f(x) = |x|/x. This function is defined for all x except x = 0.
- Step 2: Identify the left limit as x approaches 0 from the left (negative side). This means we look at values like -0.1, -0.01, etc.
- Step 3: Calculate the left limit: When x is negative, |x| = -x, so f(x) = -x/x = -1. Therefore, the left limit is -1.
- Step 4: Identify the right limit as x approaches 0 from the right (positive side). This means we look at values like 0.1, 0.01, etc.
- Step 5: Calculate the right limit: When x is positive, |x| = x, so f(x) = x/x = 1. Therefore, the right limit is 1.
- Step 6: Compare the left limit and the right limit. The left limit is -1 and the right limit is 1.
- Step 7: Since the left limit (-1) and the right limit (1) are not equal, f(x) is not continuous at x = 0.
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