2025-06-30-function-asymptotes

https://www.mathacademy.com/topics/1815 Vertical and Horizontal Asymptotes

The idea here is to use limits, what does the function look like as x approaches infinityy?

as the function approaches infinity or negative infitiy, we have a vertical asymptote? Why? Because the function is the $y$ and if it’s approaching infinitiy (positive or negative) then that’s the limit.

Horizontal Asymptotes?

Limits, Continuity, and Asymptotes

1. Why We Need Limits

Problem: Sometimes functions are undefined at certain points, but we still want to understand their behavior near those points.

Example: $f(x)=\frac{x^2-1}{x-1}$

• At $x=1$: We get $\frac{0}{0}$ (undefined)
• But we can factor: $f(x)=\frac{(x+1)(x-1)}{x-1}=x+1$ (for $x\ne 1$)
• Even though there’s a hole at $(1,2)$, the function clearly approaches $y=2$ as $x$ approaches $1$

Why this matters: Limits let us describe behavior even when the function isn’t defined at a point.

2. What Are Limits?

Definition: ${\lim }_{x\to c}f(x)=L$ means “as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$”

Key insight: We care about what happens NEAR the point, not AT the point.

Example: For $f(x)=\frac{x^2-4}{x-2}$:

• $f(1.9)=3.9$, $f(1.99)=3.99$, $f(1.999)=3.999$
• ${\lim }_{x\to 2}f(x)=4$ even though $f(2)$ is undefined

3. Continuity

Intuitive Definition: You can draw the graph without lifting your pencil.

Mathematical Definition: $f(x)$ is continuous at $x=c$ if:

1. $f(c)$ exists (function is defined there)
2. ${\lim }_{x\to c}f(x)$ exists (limit exists)
3. ${\lim }_{x\to c}f(x)=f(c)$ (limit equals function value)

Why all three conditions:

• Condition 1: No holes or undefined points
• Condition 2: Function approaches some value consistently
• Condition 3: The approached value matches the actual function value

Scope: We can check continuity at a specific point OR over an interval.

4. Vertical Asymptotes

When they occur: When the denominator of a rational function approaches zero but the numerator doesn’t.

Why they form:

• As denominator $\to 0$, the fraction $\to \pm \infty$
• Example: $f(x)=\frac{1}{x-3}$
  • As $x\to 3$: denominator $\to 0$, so $f(x)\to \pm \infty$
  • Graph shoots straight up/down (vertical line)

How to find them:

1. Factor the denominator completely
2. Set denominator = 0 and solve for $x$
3. Check that numerator ≠ 0 at those points
4. If numerator = 0 too, you have a hole, not an asymptote

Example: $f(x)=\frac{x+1}{(x-2)(x+2)}$

• Set $(x-2)(x+2)=0$
• Solutions: $x=2$ and $x=-2$
• Both are vertical asymptotes (numerator ≠ 0 at these points)

5. Horizontal Asymptotes

Purpose: Describe end behavior as $x\to \pm \infty$

Process for rational functions $f(x)=\frac{P(x)}{Q(x)}$:

1. Divide everything by highest power of $x$ in the entire expression
2. Apply: ${\lim }_{x\to \infty }\frac{1}{x^n}=0$ for any positive $n$
3. Simplify to get the limit

Three cases:

• Numerator degree > Denominator degree: No horizontal asymptote (function grows without bound)
• Numerator degree < Denominator degree: Horizontal asymptote at $y=0$
• Numerator degree = Denominator degree: Horizontal asymptote at $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

Example: $f(x)=\frac{3x^2+5x+1}{2x^2+x+7}$

1. Divide by $x^2$: $f(x)=\frac{3+\frac{5}{x}+\frac{1}{x^2}}{2+\frac{1}{x}+\frac{7}{x^2}}$

2. As $x\to \infty$: $\frac{5}{x}\to 0$, $\frac{1}{x^2}\to 0$, $\frac{1}{x}\to 0$, $\frac{7}{x^2}\to 0$

3. Result: $f(x)\to \frac{3+0+0}{2+0+0}=\frac{3}{2}$

4. Horizontal asymptote: $y=\frac{3}{2}$

6. Key Connections

Limits enable:

• Understanding function behavior near problematic points
• Defining continuity precisely
• Finding asymptotes systematically

Continuity tells us: Whether a function has any “breaks” or “jumps”

Asymptotes reveal: Long-term behavior (horizontal) and points where functions “blow up” (vertical)