Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x <

₹0.0

Question: Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }

Options:

Correct Answer: Continuous

Solution:

At x = 2, f(2) = 0 and limit from left is 0, limit from right is also 0. Hence, it is continuous.

Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }

Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }

Correct Answer: f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }

Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 - 4 for x < 2 and f(x) = 3x - 6 for x >= 2.
Step 2: Find the value of f(2) by using the second part of the function since 2 is included in that part. Calculate f(2) = 3(2) - 6.
Step 3: Calculate f(2) = 6 - 6 = 0.
Step 4: Find the limit of f(x) as x approaches 2 from the left (x < 2). Use the first part of the function: limit as x approaches 2 from the left is (2^2 - 4).
Step 5: Calculate the limit from the left: limit = 4 - 4 = 0.
Step 6: Find the limit of f(x) as x approaches 2 from the right (x >= 2). Use the second part of the function: limit as x approaches 2 from the right is (3(2) - 6).
Step 7: Calculate the limit from the right: limit = 6 - 6 = 0.
Step 8: Compare the value of f(2) with the limits from the left and right. Since f(2) = 0, limit from the left = 0, and limit from the right = 0, they are all equal.
Step 9: Conclude that the function is continuous at x = 2 because f(2) equals both limits.

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