If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x =

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Question: If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?

Options:

Correct Answer: 1

Solution:

For continuity at x = 0, we need the left limit (1) to equal k. Thus, k = 1.

If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?

If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?

Step 1: Understand that the function f(x) is defined in three parts based on the value of x: for x < 0, for x = 0, and for x > 0.
Step 2: Identify the part of the function that applies when x is less than 0, which is f(x) = x^2 + 1.
Step 3: Calculate the left limit as x approaches 0 from the left (x < 0). This means we substitute x = 0 into the left part: f(0) = 0^2 + 1 = 1.
Step 4: Identify the value of the function at x = 0, which is given as k.
Step 5: For the function to be continuous at x = 0, the left limit (1) must equal the value at x = 0 (k).
Step 6: Set the left limit equal to k: 1 = k.
Step 7: Solve for k, which gives us k = 1.

Continuity of Functions – Understanding the conditions for a function to be continuous at a point, specifically the need for the left-hand limit, right-hand limit, and the function value at that point to be equal.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, and how to evaluate limits from different sides.
Limit Evaluation – Calculating limits from the left and right to determine continuity at a specific point.