For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) c

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Question: For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) continuous at x = 1?

Options:

Correct Answer: No

Solution:

The left limit as x approaches 1 is 3, the right limit is 1, and f(1) = 3. Since the limits do not match, f(x) is discontinuous at x = 1.

For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) continuous at x = 1?

For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) continuous at x = 1?

Step 1: Identify the function f(x) and the point of interest, which is x = 1.
Step 2: Determine the left limit as x approaches 1. This means we look at the part of the function for x < 1, which is 2x + 1.
Step 3: Calculate the left limit: as x approaches 1 from the left, f(x) = 2(1) + 1 = 3.
Step 4: Determine the right limit as x approaches 1. This means we look at the part of the function for x > 1, which is x^2.
Step 5: Calculate the right limit: as x approaches 1 from the right, f(x) = (1)^2 = 1.
Step 6: Find the value of the function at x = 1. According to the function definition, f(1) = 3.
Step 7: Compare the left limit (3), right limit (1), and f(1) (3).
Step 8: Since the left limit (3) does not equal the right limit (1), the function is discontinuous at x = 1.

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 boundaries.
Limit Evaluation – Calculating left-hand and right-hand limits to determine continuity.