# How do you find the vertical, horizontal or slant asymptotes for #f(x)=(6x) / sqrt( x^2 - 3)#?

Vertical asymptotes

Horizontal asymptotes:

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To find the vertical asymptotes of the function ( f(x) = \frac{6x}{\sqrt{x^2 - 3}} ), set the denominator equal to zero and solve for ( x ). The vertical asymptotes occur where the denominator becomes zero, but the numerator does not. Thus, the vertical asymptotes occur at ( x = \pm \sqrt{3} ).

To find horizontal or slant asymptotes, we can analyze the behavior of the function as ( x ) approaches positive or negative infinity. Since the highest power of ( x ) in the numerator and denominator is 1, we can divide the leading terms to find the horizontal asymptote.

[ \lim_{x \to \infty} \frac{6x}{\sqrt{x^2 - 3}} = \lim_{x \to \infty} \frac{6x}{\sqrt{x^2(1 - \frac{3}{x^2})}} = \lim_{x \to \infty} \frac{6x}{|x|\sqrt{1 - \frac{3}{x^2}}} = \lim_{x \to \infty} \frac{6}{\sqrt{1 - 0}} = 6 ]

[ \lim_{x \to -\infty} \frac{6x}{\sqrt{x^2 - 3}} = \lim_{x \to -\infty} \frac{6x}{\sqrt{x^2(1 - \frac{3}{x^2})}} = \lim_{x \to -\infty} \frac{6x}{|x|\sqrt{1 - \frac{3}{x^2}}} = \lim_{x \to -\infty} \frac{-6}{\sqrt{1 - 0}} = -6 ]

So, the horizontal asymptote is ( y = 6 ) and ( y = -6 ).

Since the degree of the numerator is one more than the degree of the denominator, there's a slant asymptote. To find it, perform polynomial long division or synthetic division to divide ( 6x ) by ( \sqrt{x^2 - 3} ). After division, the quotient will represent the equation of the slant asymptote.

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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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