# How do you find the Vertical, Horizontal, and Oblique Asymptote given #s(t)=(8t)/sin(t)#?

Vertical:

The first graph reveals the trends, for tending towards first pair of

The second graph seems to say 'Grand New Year!'.

graph{(8x)/sin x [-128.1, 128.3, -64, 64]}

graph{(8x)/sin x [-500,500, -1000, 1000]}

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To find the vertical asymptotes, set the denominator equal to zero and solve for ( t ). For ( s(t) = \frac{8t}{\sin(t)} ), the denominator (\sin(t)) equals zero when ( t = k\pi ), where ( k ) is an integer. So, the vertical asymptotes occur at ( t = k\pi ).

Horizontal asymptotes can be determined by observing the behavior of the function as ( t ) approaches positive or negative infinity. For ( s(t) = \frac{8t}{\sin(t)} ), as ( t ) approaches infinity or negative infinity, the function behaves like ( \frac{8t}{1} ) since ( \sin(t) ) oscillates between -1 and 1. Thus, the horizontal asymptote is ( y = 8t ) or simply ( y = 8 ).

To find the oblique asymptote, perform long division or divide the numerator by the denominator. For ( s(t) = \frac{8t}{\sin(t)} ), long division would result in ( 8 + \frac{8t}{\sin(t)} ). Since ( \frac{8t}{\sin(t)} ) has vertical asymptotes at ( t = k\pi ), the oblique asymptote is the line ( y = 8t ).

So, summarizing:

- Vertical asymptotes: ( t = k\pi ) where ( k ) is an integer.
- Horizontal asymptote: ( y = 8 ).
- Oblique asymptote: ( y = 8t ).

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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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