# How do you find the asymptotes for #s(t)=t/(sin t)#?

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It has a hole (removable singularity) at

It has no horizontal or slant asymptotes.

Given:

Note that:

graph{(y-x/(sin x)) = 0 [-79.84, 80.16, -39.24, 40.76]}

graph{(y-x/(sin x))(x^2+(y-1)^2-0.002) = 0 [-2.335, 2.665, -0.49, 2.01]}

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To find the asymptotes for the function ( s(t) = \frac{t}{\sin(t)} ), we need to identify where the function approaches infinity or negative infinity.

Asymptotes can occur where the denominator of a fraction approaches zero. In this case, (\sin(t)) approaches zero at values of ( t ) such that ( t = k\pi ), where ( k ) is an integer. Therefore, we need to find the vertical asymptotes at these points.

The vertical asymptotes occur at ( t = k\pi ) where ( k ) is an integer, because at these points, the function ( s(t) ) approaches either positive or negative infinity.

Additionally, we need to check for any horizontal asymptotes. For this, we need to examine the behavior of the function as ( t ) approaches positive or negative infinity.

As ( t ) approaches positive or negative infinity, ( \sin(t) ) oscillates between -1 and 1, causing the function ( s(t) ) to oscillate without approaching any finite value. Therefore, there are no horizontal asymptotes for ( s(t) = \frac{t}{\sin(t)} ).

Thus, the asymptotes for ( s(t) = \frac{t}{\sin(t)} ) are vertical lines at ( t = k\pi ), where ( k ) is an integer.

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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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