For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) conti

For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) continuous at x = 2?

For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) continuous at x = 2?

Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 2, f(x) = 4 for x = 2, and f(x) = 2x for x > 2.
Step 2: Find the left limit as x approaches 2. This means we look at f(x) when x is just less than 2. So, we use the first part of the function: f(x) = x^2. Calculate f(2) = 2^2 = 4.
Step 3: Find the right limit as x approaches 2. This means we look at f(x) when x is just greater than 2. So, we use the third part of the function: f(x) = 2x. Calculate f(2) = 2 * 2 = 4.
Step 4: Check the value of the function at x = 2. According to the function definition, f(2) = 4.
Step 5: Compare the left limit, right limit, and the value of the function at x = 2. The left limit is 4, the right limit is 4, and f(2) is also 4.
Step 6: Since the left limit, right limit, and the value of the function at x = 2 are all equal, we conclude that f(x) is continuous at x = 2.

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 the function as it approaches a specific point.