Question

What are the vertical and horizontal asymptotes for the function f (x) = StartFraction 3 x squared Over x squared minus 4 EndFraction? horizontal asymptote: y = –2, y = 2 vertical asymptote: x = 3 horizontal asymptote: y = –4, y = 1 vertical asymptote: x = 3 horizontal asymptote: y = 3 vertical asymptote: x = –4, x = 1 horizontal asymptote: y = 3 vertical asymptote: x = –2, x = 2

Answer 1

Step-by-step explanation:

Given What are the vertical and horizontal asymptotes for the function f (x) =   3x^2 / x^2 – 4  

• We need to find the function f(x) = 3x^2 / x^2 – 4
• Now the given equation is undefined for x = 2, - 2
• When x tends to 2 the expression 3x^2 / x^2 – 4 tends to infinity from the left as well as from the right, so x = 2 is a vertical asymptote.
• When x tends to - 2 the expression 3x^2 / x^2 – 4 tends to infinity from the left as well as from the right, so x = - 2 is a vertical asymptote.
• Hence the vertical asymptotes are x = - 2, 2
• Let the rational function R (x) = p x^n / q x^m where n and m are degree of numerator and denominator.
• If n < m, then the y axis y = 0 is a horizontal asymptote.
• If n = m then horizontal asymptote is the line y = p/q
• We need to find n and m when n = 2 and m = 2
• Since n = m the horizontal asymptote is the line y = p/q where p = 3 and q = 1
• Therefore y = 3
• Now the vertical asymptotes x = - 2, 2
• Horizontal asymptotes y = 3

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Answer 2

Answer:

vertical asymptotes x = - 2, 2

Horizontal asymptotes y = 3

Step-by-step explanation:

We need to find the function f (x) = 3x ^ 2 / x ^ 2 - 4

The given equation cannot now be determined for x = 2, - 2

If x tends to 2, the expression 3x ^ 2 / x ^ 2 - 4 is probably infinite from left and right, so x = 2 is a vertical asymptote.

If x is tilted to - 2, the expression 3x ^ 2 / x ^ 2 - 4 is probably infinite from left and right, so x = - 2 is a vertical asymptote.

So the vertical asymptotes are x = - 2, 2

Let the rational function R (x) = p x ^ n / q x ^ m where n and m are the levels of the numerator and denominator.

If n <m, then the y-axis y = 0 is a horizontal asymptote.

If n = m, then the horizontal asymptote is the line y = p / q

We have to find n and m if n = 2 and m = 2

Since n = m, the horizontal asymptote is the line y = p / q, where p = 3 and q = 1

so y = 3

Now vertical asymptotes x = - 2, 2

Horizontal asymptotes y = 3

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