How do you find the asymptotes for #(4x)/(x-3)#?

Answer 1

There is a horizontal asymptote: #y = 4#

There is a vertical asymptote: #x = 3#

You can rewrite the expression.

#frac{4x}{x - 3} = 4 * frac{x}{x - 3}#
#= 4 * frac{(x - 3) + 3}{x - 3}#
#= 4 * (frac{x - 3}{x - 3} + frac{3}{x - 3})#
#= 4 * (1 + frac{3}{x - 3})#
#= 4 + frac{12}{x - 3}#

From this, you can see that

#lim_{x -> oo} frac{4x}{x - 3} = lim_{x -> oo} (4 + frac{12}{x - 3}) = 4#

Similarly,

#lim_{x -> -oo} frac{4x}{x - 3} = lim_{x -> -oo} (4 + frac{12}{x - 3}) = 4#
There is a horizontal asymptote: #y = 4#
You can also see that #x = 3# results in division by zero. When #x# approaches #3# from the left, the denominator will become infinisimally less than zero. So,
#lim_{x -> 3^-} frac{4x}{x - 3} = -oo#

Similarly,

#lim_{x -> 3^+} frac{4x}{x - 3} = oo#
There is a vertical asymptote: #x = 3#

Below is a graph for your reference. graph{(4x)/(x - 3) [-40, 40, -20, 20]}

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Answer 2

To find the asymptotes for the function ( \frac{4x}{x - 3} ), you first identify the vertical asymptote, which occurs when the denominator equals zero. Then, you check for any horizontal asymptotes by analyzing the behavior of the function as ( x ) approaches positive or negative infinity.

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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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