Question

Find the value of k, so that the following function is continuous at x = 2.

f(x) = {(x^3 + x^2 - 16x + 20)/(x - 2)^2, x ≠ 2 and
k, x = 2}

Answer 1

To make the function continuous at x = 2, k should be the value of the limit of the function as x approaches 2.

Step-by-step explanation:

To find the value of k so that the function f(x) = (x^3 + x^2 - 16x + 20) / (x - 2)^2 is continuous at x = 2, we can evaluate the limit of the function as x approaches 2. If the function approaches a finite value, then it is continuous at x = 2.

Using algebraic manipulation and factoring, we can simplify the function to (x + 5)(x - 1) / (x - 2)^2. Plugging in x = 2 to this simplified function results in a 0 in the denominator, which makes the function undefined at x = 2. Therefore, to make the function continuous at x = 2, k should be the value of the limit of the function as x approaches 2.

Let's find the limit of the function as x approaches 2. Taking the limit of the simplified function (x + 5)(x - 1) / (x - 2)^2 as x approaches 2, we get k = (2 + 5)(2 - 1) / (2 - 2)^2 = 7/0 = undefined.