Find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither.f(x) = x + 6 if x < 0 ex if 0 ≤ x ≤ 1 6 − x if x > 1x = (smaller value)continuous from the rightcontinuous from the left neitherx = (larger value)continuous from the rightcontinuous from the left neither
Question
Find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither.f(x) = x + 6 if x < 0 ex if 0 ≤ x ≤ 1 6 − x if x > 1x = (smaller value)continuous from the rightcontinuous from the left neitherx = (larger value)continuous from the rightcontinuous from the left neither
Solution 1
The function f(x) is defined by three different expressions in different intervals of x. To find the points of discontinuity, we need to check the points where the definition of the function changes, which are x = 0 and x = 1.
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At x = 0:
We check the limit of f(x) as x approaches 0 from the left and from the right.
From the left (x < 0), f(x) = x + 6, so the limit as x approaches 0 is 0 + 6 = 6.
From the right (0 ≤ x ≤ 1), f(x) = e^x, so the limit as x approaches 0 is e^0 = 1.
Since the two limits are not equal, the function is discontinuous at x = 0. It is continuous from the left, but not from the right.
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At x = 1:
We check the limit of f(x) as x approaches 1 from the left and from the right.
From the left (0 ≤ x ≤ 1), f(x) = e^x, so the limit as x approaches 1 is e^1 = e.
From the right (x > 1), f(x) = 6 - x, so the limit as x approaches 1 is 6 - 1 = 5.
Since the two limits are not equal, the function is discontinuous at x = 1. It is continuous from the left, but not from the right.
So, the function f(x) is discontinuous at x = 0 and x = 1. It is continuous from the left at these points, but not from the right.
Solution 2
The function f(x) is defined by three different expressions in three different intervals of x. Therefore, the points of discontinuity can only occur at the boundaries of these intervals, which are x = 0 and x = 1.
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At x = 0:
To check if the function is continuous from the right at x = 0, we need to check if the limit of f(x) as x approaches 0 from the right is equal to f(0).
The function is defined by e^x for 0 ≤ x ≤ 1, so f(0) = e^0 = 1.
The limit of f(x) as x approaches 0 from the right is also 1, because e^x approaches 1 as x approaches 0 from the right.
Therefore, the function is continuous from the right at x = 0.
To check if the
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