How to Evaluate Limits at Infinity of Rational Functions

Understanding Limits at Infinity for Rational Functions

• 💡 When evaluating limits at infinity for a bottom-heavy rational function (degree of numerator < degree of denominator), the limit will always approach zero.
• 🔬 To formally show work, divide both the numerator and denominator by the highest power of x in the denominator (e.g., x^2), then apply the rule that the limit of 1/x^n as x approaches infinity is zero.

Limits When Degrees of Numerator and Denominator are Equal

• 🎯 If the degree of the numerator is equal to the degree of the denominator, the limit at infinity is the ratio of the leading coefficients.
• 🔑 For example, in a function where the highest power is x^3 in both numerator and denominator, the limit is the coefficient of the x^3 term in the numerator divided by the coefficient of the x^3 term in the denominator.
• 📈 To show work, divide all terms by the highest power of x (e.g., x^3) and apply the limit rule for 1/x^n approaching zero.

Limits at Infinity for Top-Heavy Functions

• ⚠️ When the degree of the numerator exceeds the degree of the denominator (a top-heavy function), the limit will approach either positive or negative infinity.
• ⚡ To quickly determine the limit, simplify the function by considering only the most significant terms (highest powers of x) in the numerator and denominator.
• 🚀 For formal work, divide the numerator and denominator by the highest power of x found in the denominator, even if the function is top-heavy.
• 🧠 The limit of 1/x as x approaches negative infinity is also zero, similar to when x approaches positive infinity.