What are the vertical and horizontal asymptotes for the function f(x) = 3x²/x² - 4

Solution:

Horizontal asymptotes are nothing but the lines parallel to x - axis which exists when fꞌ(x) = 0

And vertical asymptotes are nothing but the lines parallel to y - axis which exists when fꞌ(x) = Not defined.

Consider f(x)=3x²/x² - 4

Differentiate w.r.to x

fꞌ(x) = {(x² - 4) (6x ) - 3x² (2x)}/ (x² - 4)²

Now for fꞌ(x) = 0

⇒ (x² - 4) (6x ) - 3x² (2x) = 0

⇒ 2x { 3x² - 12x - 3x² } = 0

⇒ x = 0

∴ Horizontal asymptotes is x = 0 ⇒ y = 0 which is x - axis

Now for fꞌ(x) =∞

⇒(x² - 4)² = 0

⇒ x² = 4

⇒ x = ± 2

∴ Vertical asymptotes are x = 2 and x = -2

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What are the vertical and horizontal asymptotes for the function f(x) = 3x²/x² - 4

Summary:

Horizontal asymptotes is x = 0 ⇒ y = 0 which is x - axis, Vertical asymptotes are x = 2 and x = -2.