If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 }, what value of k makes f continuous at x = 0?

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Q: If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 }, what value of k makes f continuous at x = 0?

Solution: To be continuous at x = 0, k must equal the limit from the left, which is 1.

Steps: 7

Step 1: Understand that f(x) is a piecewise function, which means it has different expressions for different values of x.
Step 2: Identify the three parts of the function: f(x) = x^2 + 1 when x < 0, f(x) = k when x = 0, and f(x) = 2x + 1 when x > 0.
Step 3: To find the value of k that makes f continuous at x = 0, we need to ensure that the left-hand limit (as x approaches 0 from the left) equals the right-hand limit (as x approaches 0 from the right) and also equals f(0).
Step 4: Calculate the left-hand limit: as x approaches 0 from the left (x < 0), use the expression x^2 + 1. So, limit as x approaches 0 from the left is 0^2 + 1 = 1.
Step 5: Calculate the right-hand limit: as x approaches 0 from the right (x > 0), use the expression 2x + 1. So, limit as x approaches 0 from the right is 2(0) + 1 = 1.
Step 6: Since both limits (from the left and right) equal 1, for f(x) to be continuous at x = 0, we need k to also equal 1.
Step 7: Therefore, the value of k that makes f continuous at x = 0 is k = 1.