Question

Check the continuity as x approaches 2 for the function (f(x)=x²-4/2-|x|).

a) Continuous at x = 2
b) Discontinuous at x = 2
c) Continuous for x < 2 but not at x = 2
d) Continuous for x > 2 but not at x = 2

Answer 1

To check the continuity as x approaches 2 for the function (f(x)=x²-4/2-|x|), evaluate the limit from the left and right sides of x=2 and compare with the value of the function at 2. The right-side limit is undefined, so the function is discontinuous at x=2.

Step-by-step explanation:

To check the continuity as x approaches 2 for the function f(x) = (x^2 - 4) / (2 - |x|), we need to evaluate the limit of the function as x approaches 2 from both the left and the right sides, and then compare these limits to the value of the function at 2. Let's start with the limit from the left side (x < 2):

lim(x ← 2-) (x^2 - 4) / (2 - |x|) = lim(x ← 2-) (x^2 - 4) / (2 - (-x)) = lim(x ← 2-) (x^2 - 4) / (2 + x) = (-4) / 4 = -1

Now, let's calculate the limit from the right side (x > 2):

lim(x → 2+) (x^2 - 4) / (2 - |x|) = lim(x → 2+) (x^2 - 4) / (2 - x) = lim(x → 2+) (x^2 - 4) / (2 - x) = (-4) / 0

Since the right-side limit is undefined, the function is discontinuous at x = 2. Therefore, the correct answer is option b) Discontinuous at x = 2.