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Answer :
To find the limit of the piecewise function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 1, we need to consider the behavior of the function from both sides of [tex]\( x = 1 \)[/tex].
The function is defined as:
- [tex]\( f(x) = z + 3 \)[/tex] for [tex]\( z < 1 \)[/tex]
- [tex]\( f(x) = -2z + 7 \)[/tex] for [tex]\( z > 1 \)[/tex]
Step 1: Find the left-hand limit as [tex]\( z \)[/tex] approaches 1 from the left ([tex]\( z < 1 \)[/tex]):
For values of [tex]\( z \)[/tex] that are slightly less than 1, we use the expression [tex]\( z + 3 \)[/tex].
- If we plug [tex]\( z = 1 \)[/tex] into [tex]\( z + 3 \)[/tex], we get:
[tex]\[
1 + 3 = 4
\][/tex]
Step 2: Find the right-hand limit as [tex]\( z \)[/tex] approaches 1 from the right ([tex]\( z > 1 \)[/tex]):
For values of [tex]\( z \)[/tex] that are slightly greater than 1, we use the expression [tex]\(-2z + 7\)[/tex].
- If we plug [tex]\( z = 1 \)[/tex] into [tex]\(-2z + 7\)[/tex], we get:
[tex]\[
-2(1) + 7 = -2 + 7 = 5
\][/tex]
Conclusion:
The left-hand limit is 4, and the right-hand limit is 5. Since the left-hand limit and the right-hand limit are not equal, the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 1 does not exist.
The correct answer is (d) none.
The function is defined as:
- [tex]\( f(x) = z + 3 \)[/tex] for [tex]\( z < 1 \)[/tex]
- [tex]\( f(x) = -2z + 7 \)[/tex] for [tex]\( z > 1 \)[/tex]
Step 1: Find the left-hand limit as [tex]\( z \)[/tex] approaches 1 from the left ([tex]\( z < 1 \)[/tex]):
For values of [tex]\( z \)[/tex] that are slightly less than 1, we use the expression [tex]\( z + 3 \)[/tex].
- If we plug [tex]\( z = 1 \)[/tex] into [tex]\( z + 3 \)[/tex], we get:
[tex]\[
1 + 3 = 4
\][/tex]
Step 2: Find the right-hand limit as [tex]\( z \)[/tex] approaches 1 from the right ([tex]\( z > 1 \)[/tex]):
For values of [tex]\( z \)[/tex] that are slightly greater than 1, we use the expression [tex]\(-2z + 7\)[/tex].
- If we plug [tex]\( z = 1 \)[/tex] into [tex]\(-2z + 7\)[/tex], we get:
[tex]\[
-2(1) + 7 = -2 + 7 = 5
\][/tex]
Conclusion:
The left-hand limit is 4, and the right-hand limit is 5. Since the left-hand limit and the right-hand limit are not equal, the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 1 does not exist.
The correct answer is (d) none.
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