If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuo
Practice Questions
Q1
If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuous at x = 0, k must be:
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Questions & Step-by-Step Solutions
If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuous at x = 0, k must be:
Step 1: Understand that for a function to be continuous at a point, the value of the function at that point must equal the limit of the function as it approaches that point.
Step 2: Identify the point we are interested in, which is x = 0.
Step 3: Calculate the limit of f(x) as x approaches 0 from the left (x < 0). This means we use the part of the function f(x) = x^2 + 1.
Step 4: Substitute 0 into the left-side function: f(0) = 0^2 + 1 = 1.
Step 5: Now calculate the limit of f(x) as x approaches 0 from the right (x > 0). This means we use the part of the function f(x) = 2x.
Step 6: Substitute 0 into the right-side function: f(0) = 2 * 0 = 0.
Step 7: For the function to be continuous at x = 0, the left limit (1) must equal the right limit (0) and also equal k (the value of the function at x = 0).
Step 8: Since the left limit is 1, we set k = 1 to ensure continuity.
Piecewise Functions – Understanding how piecewise functions are defined and how to evaluate them at specific points.
Limits and Continuity – Applying the concept of limits to determine the continuity of a function at a specific point.
