How do you find the slant asymptote of #y=(x^3)/((x^2)-3)#?

Answer 1

The slant asymptote is: #y=x#

Given:

#y = x^3/(x^2-3)#

Note that:

#x^3/(x^2-3) = (x^3-3x+3x)/(x^2-3)#
#color(white)(x^3/(x^2-3)) = (x(x^2-3)+3x)/(x^2-3)#
#color(white)(x^3/(x^2-3)) = x+(3x)/(x^2-3)#

and:

#lim_(x->+-oo) (3x)/(x^2-3) = lim_(x->+-oo) 3/(x-3/x) = lim_(x->+-oo)3/x = 0#

So:

#y = x^3/(x^2-3)#
is asymptotic to #y=x#
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Answer 2

To find the slant asymptote of the function ( y = \frac{x^3}{x^2 - 3} ), follow these steps:

  1. Perform polynomial long division to divide ( x^3 ) by ( x^2 - 3 ). This gives you the quotient and remainder.

  2. The quotient obtained from the division is the equation of the slant asymptote.

Let's perform the division:

[ \frac{x^3}{x^2 - 3} ]

[ = x \times \frac{x^2}{x^2 - 3} ]

[ = x \times \left(1 + \frac{3}{x^2 - 3}\right) ]

Now, divide ( x^2 ) by ( x^2 - 3 ):

[ = x \times \left(1 + \frac{3}{x^2 - 3}\right) ]

[ = x + \frac{3x}{x^2 - 3} ]

The quotient of the division is ( x + \frac{3x}{x^2 - 3} ), which represents the equation of the slant asymptote.

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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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