For what values of x, if any, does #f(x) = 1/((4x+9)cos(pi/2+(4pi)/x) # have vertical asymptotes?

Answer 1

Asymptote is vertical at

#x=-9/4, 4/k#

Where #k=0, \pm1, \pm2, \pm3, \ldots#

The given function:

#f(x)=1/{(4x+9)\cos(\pi/2+{4\pi}/x)}#
#f(x)=1/{(4x+9)\sin({4\pi}/x)}#

Above function will have vertical asymptotes when denominator becomes equal to zero i.e.

#(4x+9)\sin({4\pi}/x)=0#
#4x+9=0, \ \ or\ \ \sin({4\pi}/x)=0#
#x=-9/4, \ \ or\ \ \ {4\pi}/x=k\pi#
#x=-9/4\ \ \or \ \ \ x=4/k#
Where, #k# is any integer hence we get the set of points where asymptote is vertical
#x=-9/4, 4/k#
Where #k=0, \pm1, \pm2, \pm3, \ldots#
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Answer 2

The function f(x) has vertical asymptotes at values of x where the denominator of the function becomes zero. In this case, the denominator is (4x+9)cos(pi/2+(4pi)/x). To find the values of x that make the denominator zero, we set it equal to zero and solve for x. However, it is important to note that the cosine function has a period of 2π, so we need to consider values of x that make the argument of the cosine function equal to odd multiples of π/2. Therefore, the values of x that make the denominator zero and result in vertical asymptotes are x = -9/4 and x = -9/4 + (2n+1)π/4, where n is an integer.

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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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