What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x
Practice Questions
Q1
What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?
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Questions & Step-by-Step Solutions
What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?
Step 1: Understand that we want the function f(x) to be continuous at x = 0.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = kx for x < 0 and f(x) = x^2 + 1 for x >= 0.
Step 4: Calculate the value of the function at x = 0 using the right-hand side: f(0) = 0^2 + 1 = 1.
Step 5: Calculate the left-hand limit as x approaches 0 from the left (x < 0): limit as x approaches 0 of kx = k(0) = 0.
Step 6: Set the left-hand limit equal to the value of the function at x = 0: k(0) = 1.
Step 7: Solve for k: Since k(0) = 0, we need to set 0 = 1, which is not possible. Instead, we need to find k such that k(0) = 1.
Step 8: Since we want the left-hand limit (0) to equal the right-hand limit (1), we find that k must be 1.
