Learn how to find horizontal asymptotes of rational functions by analyzing the degrees of numerator and denominator polynomials | Step-by-Step Solution

Finding Horizontal Asymptotes in Rational Functions

What We're Solving:

We're learning how to find horizontal asymptotes of rational functions by examining the relationship between the degrees and leading coefficients of the numerator and denominator polynomials.

The Approach:

A horizontal asymptote represents the "end behavior" of a function - where the graph heads as x approaches positive or negative infinity. For rational functions, this behavior is determined by which polynomial (top or bottom) "grows faster" and by how much.

Step-by-Step Solution:

Step 1: Identify the degrees

• Find the degree of the numerator polynomial (highest power of x on top)
• Find the degree of the denominator polynomial (highest power of x on bottom)
• Compare these degrees - this comparison tells us everything!

Step 2: Apply the three rules based on degree comparison

Case 1: Degree of numerator < Degree of denominator

• The horizontal asymptote is y = 0 (the x-axis)
• The denominator grows much faster, making the whole fraction approach zero
• Example: f(x) = (3x + 1)/(x² + 2) → y = 0

Case 2: Degree of numerator = Degree of denominator

• The horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator)
• Both polynomials grow at the same rate, so their ratio approaches the ratio of their leading terms
• Example: f(x) = (4x² + 3x)/(2x² + 5) → y = 4/2 = 2

Case 3: Degree of numerator > Degree of denominator

• There is NO horizontal asymptote (the function grows without bound)
• The numerator grows faster, so the fraction keeps getting larger
• Example: f(x) = (x³ + 1)/(x² + 2) → No horizontal asymptote

Step 3: Double-check your work

• Verify you identified degrees correctly
• Make sure you used the right leading coefficients (the numbers in front of the highest degree terms)

The Answer:

The horizontal asymptote depends entirely on comparing degrees:

• Bottom-heavy (denominator degree > numerator degree) → y = 0
• Equal weight (same degrees) → y = ratio of leading coefficients
• Top-heavy (numerator degree > denominator degree) → No horizontal asymptote

Memory Tip:

Remember "Bottom wins, tie game, top wins":

• Bottom wins (higher degree): asymptote at y = 0
• Tie game (equal degrees): asymptote at y = coefficient ratio
• Top wins (higher degree): no horizontal asymptote

You're essentially asking "Who's stronger?" between the top and bottom of the fraction when x gets huge! 🎯