QuestionIs the function continuous at , where if and ?
Studdy Solution
STEP 1
Assumptions1. The function is defined as for all and for .
. The value of is .
3. We are checking for continuity at .
STEP 2
A function is continuous at a point if the following three conditions are met1. is defined.
2. exists.
. .
We will check each of these conditions.
STEP 3
First, we check if is defined. From the definition of the function, we know that .
STEP 4
Next, we need to calculate the limit of as approaches from both the left and the right. If both of these limits exist and are equal, then exists.
STEP 5
We calculate the left-hand limit, which is .
STEP 6
Since is approaching from the left (i.e., ), the expression inside the absolute value, , is negative. Therefore, .
STEP 7
Now, we substitute into the expression to calculate the left-hand limit.
STEP 8
Next, we calculate the right-hand limit, which is .
STEP 9
Since is approaching from the right (i.e., ), the expression inside the absolute value, , is positive. Therefore, .
STEP 10
Now, we substitute into the expression to calculate the right-hand limit.
STEP 11
Since the left-hand limit () and the right-hand limit ($$) are not equal, $\lim_{x \to c} f(x)$ does not exist.
STEP 12
Therefore, the function is not continuous at , because it does not meet all three conditions for continuity at that point.
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