For what values of x, if any, does #f(x) = 1/((12x-9)sin(pi+(3pi)/x) # have vertical asymptotes?

Answer 1

#x=0,3/k,kinZZ# (where #k# is an integer)

Note that for a rational function such as the given function #f#, there will be a rational function whenever its denominator equals #0#.
So, there is a vertical asymptote whenever #(12x-9)sin(pi+(3pi)/x)=0#.

We can split this into two parts:

#{(12x-9=0),(sin(pi+(3pi)/x)=0):}#
The first can be solved to show that #x=9/12=3/4#.
The second is a little more difficult: note that #sin(x)=0# when #x=0,pi,2pi#, and so on. This can be written as #x=kpi#, where #kinZZ#, which means #k# is an integer.

Thus:

#pi+(3pi)/x=kpi" "" "," " "kinZZ#
Subtracting #pi#, we see that the right hand side remains #kpi#, because some integer multiple of #pi# minus #pi# is still an integer multiple of #pi#.
#(3pi)/x=kpi" "" "," " " "kinZZ#

Rearranging:

#3pi=x(kpi)" "" "," "" "kinZZ#
#x=(3pi)/(kpi)=3/k" "" "," "" "kinZZ#
Note that the previous solution from #12x-9=0# gave #x=3/4#, which is included in the solution #x=3/k,kinZZ#.
Furthermore, since #(3pi)/x# is included within the sine function, there will be a vertical asymptote at #x=0#.
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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 (12x-9)sin(pi+(3pi)/x). To find the values of x that make the denominator zero, we set it equal to zero and solve for x. However, we need to be careful because the sine function has periodic behavior.

Setting the denominator equal to zero: (12x-9)sin(pi+(3pi)/x) = 0

To find the values of x that make the sine function zero, we have: sin(pi+(3pi)/x) = 0

The sine function is zero at integer multiples of pi. So, we can write: pi + (3pi)/x = n*pi, where n is an integer

Simplifying the equation: (3pi)/x = (n-1)*pi 3/x = n-1

Solving for x: x = 3/(n-1)

Therefore, the function f(x) has vertical asymptotes at x = 3/(n-1), where n is an integer except when n = 1.

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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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