Determine if the function 𝑓 ( 𝑥 ) = { cos ⁡ ( 𝑥 ) 𝑥 0 is continuous at 𝑥 = 0 . If it is not continuous, determine the correct reason.

Question

Determine if the function 
𝑓
(
𝑥
)
=
{
cos
⁡
(
𝑥
)
	
𝑥
<
0

0
	
𝑥
=
0

1
−
𝑥
2
	
𝑥
>
0
 is continuous at 
𝑥
=
0
. If it is not continuous, determine the correct reason.

BlackTom AI · Accepted Answer

Let's lay out the function piece by piece to inspect its behavior near x = 0.
- For x < 0, f(x) = cos(x). As x approaches 0 from the left, cos(x) approaches cos(0) = 1, so the left-hand limit is 1.
- For x > 0, f(x) = 1 − x^2. As x approaches 0 from the right, 1 − x^2 approaches 1, so the right-hand limit is also 1.
- At x = 0, f(0) is defined to be 0.
Now, compare these observations to the criteria for continuity at a point. A function is continuous at x = 0 if the limit as x approaches 0 exists and equals f(0).
Option A: 'f is continuous at x = 0.' This is contradicted by the fact that f(0) = 0 while the limit from both sides is 1, so this option is incorrect.
Option B: 'f is not continuous at x = 0 because lim x→0 f(x) does not exist.' Here we see the left- and right-hand limits both exist and equal 1, so the limit does exist. Therefore, this statement is false.
Option C: 'f is not continuous at x = 0 because f(0) does not exist.' The definition assigns f(0) = 0, so f(0) does exist. This makes this option incorrect as well.
Option D: 'f is not continuous at x = 0 because lim x→0 f(x) ≠ f(0).' This captures the essential reason: the limit exists and equals 1, but f(0) = 0, so the limit does not equal the function value at 0, violating the condition for continuity. This option correctly identifies the discontinuity by comparing the limit to the actual value at the point.
In summary, the only statement that accurately describes why the function is not continuous at x = 0 is the one asserting that the limit as x approaches 0 does not equal f(0).

MATH 2A LEC A: CALCULU... Quiz 3 (Multiple Choice)

Determine if the function 𝑓 ( 𝑥 ) = { cos ⁡ ( 𝑥 ) 𝑥 < 0 0 𝑥 = 0 1 − 𝑥 2 𝑥 > 0 is continuous at 𝑥 = 0 . If it is not continuous, determine the correct reason.

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