How do you find the x values at which #f(x)=cos((pix)/2)# is not continuous, which of the discontinuities are removable?

Answer 1

The function is continuous everywhere.

The function #g(x) = (pix)/2# is linear. It is continuous at every #x#.

The cosine function is continuous everywhere.

The composition of continuous functions is continuous.

#f# is the composition of #cos(x)# with #(pix)/2#, so #f# is continuous everywhere.
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Answer 2

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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Answer from HIX Tutor

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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