Determine if the function f(x) = { x^2, x < 0; 1/x, x > 0 } is continuous
Practice Questions
Q1
Determine if the function f(x) = { x^2, x < 0; 1/x, x > 0 } is continuous at x = 0.
Yes
No
Depends on limit
None of the above
Questions & Step-by-Step Solutions
Determine if the function f(x) = { x^2, x < 0; 1/x, x > 0 } is continuous at x = 0.
Correct Answer: f(x) is not continuous at x = 0.
Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 when x is less than 0, and f(x) = 1/x when x is greater than 0.
Step 2: To check continuity at x = 0, we need to find the left limit (as x approaches 0 from the left) and the right limit (as x approaches 0 from the right).
Step 3: Calculate the left limit: As x approaches 0 from the left (x < 0), f(x) = x^2. The limit is lim (x -> 0-) x^2 = 0.
Step 4: Calculate the right limit: As x approaches 0 from the right (x > 0), f(x) = 1/x. The limit is lim (x -> 0+) 1/x, which is undefined (it goes to infinity).
Step 5: Compare the left limit and the right limit. The left limit is 0 and the right limit is undefined.
Step 6: Since the left limit and right limit are not equal, the function 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.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at points where the definition changes.
Limits – Calculating left-hand and right-hand limits to determine the behavior of a function as it approaches a specific point.
