Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }
Correct Answer: f(x) is continuous at x = 2.
- Step 1: Identify the function f(x) and the point of interest, which is x = 2.
- Step 2: Determine the value of the function at x = 2. Here, f(2) = 4.
- Step 3: Calculate the left limit as x approaches 2. For x < 2, f(x) = x^2. So, left limit = limit as x approaches 2 from the left of x^2 = 2^2 = 4.
- Step 4: Calculate the right limit as x approaches 2. For x > 2, f(x) = 2x. So, right limit = limit as x approaches 2 from the right of 2x = 2*2 = 4.
- Step 5: Compare the left limit, right limit, and f(2). All three values are equal to 4.
- Step 6: Since the left limit, right limit, and f(2) are all equal, conclude that f(x) is continuous at x = 2.
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