For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) co

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Question: For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) continuous at x = 3?

Options:

Correct Answer: Yes

Solution:

The left limit as x approaches 3 is 9, the right limit is also 9, and f(3) = 9. Therefore, f(x) is continuous at x = 3.

For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) continuous at x = 3?

For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) continuous at x = 3?

Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 3, f(x) = 9 for x = 3, and f(x) = x + 3 for x > 3.
Step 2: Calculate the left limit as x approaches 3. This means we look at f(x) when x is just a little less than 3. For x < 3, f(x) = x^2. So, we find the limit: lim (x -> 3-) f(x) = 3^2 = 9.
Step 3: Calculate the right limit as x approaches 3. This means we look at f(x) when x is just a little more than 3. For x > 3, f(x) = x + 3. So, we find the limit: lim (x -> 3+) f(x) = 3 + 3 = 6.
Step 4: Check the value of the function at x = 3. According to the function definition, f(3) = 9.
Step 5: Compare the left limit, right limit, and the value of the function at x = 3. The left limit is 9, the right limit is 6, and f(3) is 9.
Step 6: Since the left limit (9) does not equal the right limit (6), f(x) is not continuous at x = 3.

Continuity of Functions – Understanding the definition of continuity at a point, which requires that the left limit, right limit, and the function value at that point are all equal.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at the point where the definition changes.
Limits – Calculating left-hand and right-hand limits to determine the behavior of a function as it approaches a specific point.