Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 2, x > 2 is continuous at x = 2.
Practice Questions
1 question
Q1
Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 2, x > 2 is continuous at x = 2.
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For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right and equal to f(2). Thus, k = 4.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 2, x > 2 is continuous at x = 2.
Solution: For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right and equal to f(2). Thus, k = 4.
Steps: 7
Step 1: Understand that for a function to be continuous at a point, the limit from the left side and the limit from the right side must be equal to the function's value at that point.
Step 2: Identify the function f(x) and the point of interest, which is x = 2.
Step 3: Calculate the limit of f(x) as x approaches 2 from the left (x < 2). This means using the part of the function x^2 - 4. So, find the limit: limit as x approaches 2 from the left of (x^2 - 4).
Step 4: Substitute 2 into the left-side function: (2^2 - 4) = 4 - 4 = 0.
Step 5: Calculate the limit of f(x) as x approaches 2 from the right (x > 2). This means using the part of the function 3x - 2. So, find the limit: limit as x approaches 2 from the right of (3x - 2).
Step 6: Substitute 2 into the right-side function: (3*2 - 2) = 6 - 2 = 4.
Step 7: For the function to be continuous at x = 2, the left limit (0) must equal the right limit (4) and also equal f(2), which is k. Therefore, we set k equal to the right limit: k = 4.
