QuestionFind all values of where exists for the piecewise function defined above.
Studdy Solution
STEP 1
Assumptions1. The function is defined piecewise for as follows - for - for - for . We are asked to find all values of at which the limit of as approaches exists.
STEP 2
For a limit to exist at a certain point , the left-hand limit and the right-hand limit at that point must be equal. That is, .
STEP 3
Let's first consider the points where the definition of the function changes, namely , , and .
STEP 4
For , we have to check if .
STEP 5
Calculate the left-hand limit at using the definition of for .
STEP 6
Calculate the right-hand limit at using the definition of for .
STEP 7
Notice that the right-hand limit at does not exist because the function is not defined for in the second piece. Therefore, the limit at does not exist.
STEP 8
Now, let's consider . We have to check if .
STEP 9
Calculate the left-hand limit at using the definition of for .
STEP 10
Calculate the right-hand limit at using the definition of for .
STEP 11
Notice that the left-hand limit at does not exist because the function is not defined for in the first piece. Therefore, the limit at does not exist.
STEP 12
Now, let's consider . We have to check if .
STEP 13
Calculate the left-hand limit at using the definition of for .
STEP 14
Calculate the right-hand limit at using the definition of for .
STEP 15
Calculate the values of the left-hand and right-hand limits at .
STEP 16
Since the left-hand limit equals the right-hand limit at , the limit at exists.
STEP 17
For all other values of not equal to , , or , the function is continuous and therefore the limit exists.
STEP 18
Therefore, the values of at which exists are all such that , and the limit exists at .
STEP 19
Expressing this in interval notation, the values of at which exists are .
The values of at which the limit of as approaches exists are .
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