What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?

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Q: What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?

Solution: Setting the left limit (k(1) = k) equal to the right limit (1 + 1 = 2), we find k = 2.

Steps: 7

Step 1: Identify the function f(x) which has different expressions based on the value of x.
Step 2: Recognize that we need to check the continuity of f(x) at x = 1.
Step 3: Find the left limit as x approaches 1 from the left (x < 1). This is given by kx. So, the left limit is k(1) = k.
Step 4: Find the value of the function at x = 1. According to the function, f(1) = 2.
Step 5: Find the right limit as x approaches 1 from the right (x > 1). This is given by x + 1. So, the right limit is 1 + 1 = 2.
Step 6: Set the left limit equal to the right limit to ensure continuity. This means k must equal 2.
Step 7: Conclude that the value of k that makes the function continuous at x = 1 is k = 2.