Asymptotes — Definition and Types

Asymptote is a straight line that a curve approaches closer and closer but never actually touches or crosses as the variable tends to infinity or some specific value.

Types of Asymptotes:

There are three types of asymptotes:

• Vertical Asymptote: A vertical line $x = a$ where the function approaches $\pm\infty$ as $x \to a$. It occurs where the denominator becomes zero.

• Horizontal Asymptote: A horizontal line $y = b$ where the function approaches a finite value $b$ as $x \to \pm\infty$. It describes the end behavior of the function.

• Oblique (Slant) Asymptote: A slanted line $y = mx + c$ that the curve approaches as $x \to \pm\infty$. It occurs when the degree of the numerator is exactly one more than the degree of the denominator.

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Find horizontal and vertical asymptotes of: $f(x) = \dfrac{-8}{x^2 - 4}$

Vertical Asymptotes:

Set the denominator equal to zero:

$$x^2 - 4 = 0$$

$$(x - 2)(x + 2) = 0$$

$$x = 2 \quad \text{and} \quad x = -2$$

Check that numerator $\neq 0$ at these points: $-8 \neq 0$ ✓

Vertical Asymptotes: $x = 2$ and $x = -2$

Horizontal Asymptotes:

Compare the degree of numerator and denominator:

• Degree of numerator = 0 (constant $-8$)
• Degree of denominator = 2 ($x^2 - 4$)

Since degree of numerator < degree of denominator:

$$\lim_{x \to \pm\infty} \frac{-8}{x^2 - 4} = 0$$

Horizontal Asymptote: $y = 0$ (the x-axis)

Conclusion: The function has two vertical asymptotes at $x = 2$ and $x = -2$, and one horizontal asymptote at $y = 0$.

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No, oblique (slant) asymptote does not exist because the degree of the numerator (0) is not exactly one more than the degree of the denominator (2).