# For what values of x, if any, does #f(x) = 1/((12x+4)sin(pi+(6pi)/x) # have vertical asymptotes?

Vertical asymptotes :

Horizontal asymptote :

The asymptotes are given by

The horizontal asymptote is revealed by

The horizontal space between consecutive vertical asymptotes

You can study the second graph, for shape near the exclusive

I have used ad hoc ( for the purpose ) scales, for clarity.

graph{(4y(3x+1)sin(6pi/x)+1)(x-6-.01y)(x+6+.01y)=0 [-16, 16, -.5, .5]}

graph{(4y(3x+1)sin(6pi/x)+1)(x+.333-.00001y)=0 [-.4 -.0,-10, 10]}

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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+4)sin(pi+(6pi)/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 sin(pi+(6pi)/x) will never be zero, as the sine function oscillates between -1 and 1. Therefore, there are no values of x that make the denominator zero, and consequently, there are no vertical asymptotes for the function f(x).

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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.

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