Limit Lab

THE WHY BEHIND
ASYMPTOTES

Connecting Calculus Limits to Horizontal Asymptotes

UNIT 4: LIMITS AT INFINITY

Warm-up: The Vanishing Fraction

Evaluate the expression 10 / x for the following values of x:

x = 10 ?

x = 100 ?

x = 1,000 ?

Big Question:

"As x approaches infinity, what value does the entire fraction approach?"

\[ \lim_{x \to \infty} \frac{10}{x} = \dots \]

Video Investigation

Horizontal Asymptotes in Rational Functions

10:00 Duration

Watch for:

Can a graph cross a horizontal asymptote?
What is "Case 2" (Equal Degrees)?
How do we find the "Ratio of Coefficients"?

Case 2: Equal Degrees

The Shortcut Rule:

If degree of Num = degree of Denom,
Asymptote is y = a / b

Example 1

\[ f(x) = \frac{10x^4 + 3}{2x^4 - 8} \]

Degrees: 4 and 4

Ratio: 10 / 2 = 5

Asymptote: y = 5

Warning!

Example 2

\[ g(x) = \frac{7x^2 + 5}{x - 3x^2} \]

Coefficients: 7 and -3

Asymptote: y = -7/3

Look for the HIGHEST power, not the first term!

The "Calculus Way"

Why does the rule work?

Calculus doesn't just "look" at the highest powers. It evaluates the limit using algebra:

The Strategy:

Divide EVERY term in the function by the highest power of x found in the denominator.

The Transformation:

\[ \frac{5x + 3}{x - 2} \]

Divide by x:

\[ \frac{\frac{5x}{x} + \frac{3}{x}}{\frac{x}{x} - \frac{2}{x}} \]

Simplify:

\[ \frac{5 + 0}{1 - 0} = 5 \]

Peer Explain

Explain to your partner why x² / x³ must approach 0 as x → ∞.

Focus on:

The "Heavy Bottom" logic
What happens to 1/x as x grows?

The Limit Lab

Limit Lab Teacher Guide

Limit Lab Slides

The Why Worksheet

The Why Answer Key

Asymptote Anchor Chart

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