Question: Is the function f(x) = |x|/x continuous at x = 0?
Options:
Yes
No
Depends on direction
None of the above
Correct Answer: No
Solution:
The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.
Is the function f(x) = |x|/x continuous at x = 0?
Practice Questions
Q1
Is the function f(x) = |x|/x continuous at x = 0?
Yes
No
Depends on direction
None of the above
Questions & Step-by-Step Solutions
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.
Continuity of Functions – Understanding the definition of continuity at a point, which requires that the left-hand limit, right-hand limit, and the function value at that point are all equal.
Limits – Evaluating one-sided limits to determine the behavior of a function as it approaches a specific point from either side.
Piecewise Functions – Recognizing that the function f(x) = |x|/x is piecewise defined and behaves differently for positive and negative values of x.
Soulshift Feedback×
On a scale of 0–10, how likely are you to recommend
The Soulshift Academy?
