How do you find the x values at which #f(x)=cos((pix)/2)# is not continuous, which of the discontinuities are removable?
The function is continuous everywhere.
The cosine function is continuous everywhere.
The composition of continuous functions is continuous.
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To find the x values at which f(x) = cos((πx)/2) is not continuous, we need to identify the points where the function has a discontinuity.
The function f(x) = cos((πx)/2) is not continuous when the argument of the cosine function, (πx)/2, is equal to odd multiples of π/2.
Setting (πx)/2 equal to odd multiples of π/2, we have:
(πx)/2 = (2n + 1)(π/2), where n is an integer.
Simplifying the equation, we get:
x = 2n + 1, where n is an integer.
Therefore, the x values at which f(x) = cos((πx)/2) is not continuous are given by x = 2n + 1, where n is an integer.
Now, to determine which of these discontinuities are removable, we need to check if the limit of the function exists at those points. If the limit exists, the discontinuity is removable; otherwise, it is not removable.
Taking the limit of f(x) as x approaches each of the x values, x = 2n + 1, we find that the limit exists for all these points. Therefore, all the discontinuities of f(x) = cos((πx)/2) are removable.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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