AP Calculus BC 1.15 Connecting Limits at Infinity and Horizontal Asymptotes Study Notes - New Syllabus
AP Calculus BC 1.15 Connecting Limits at Infinity and Horizontal Asymptotes Study Notes- New syllabus
AP Calculus BC 1.15 Connecting Limits at Infinity and Horizontal Asymptotes Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
Interpret the behavior of functions using limits involving infinity.
Key Concepts:
- Connecting Limits at Infinity and Horizontal Asymptotes
Connecting Limits at Infinity and Horizontal Asymptotes
Connecting Limits at Infinity and Horizontal Asymptotes
A function \( f(x) \) has a horizontal asymptote if the value of \( f(x) \) approaches a finite number as \( x \) approaches \( \infty \) or \( -\infty \).
This is expressed as:
\( \lim_{x \to \infty} f(x) = L \) or \( \lim_{x \to -\infty} f(x) = L \).
The line \( y = L \) is the horizontal asymptote.
Key Points:
- Rational functions often have horizontal asymptotes determined by comparing degrees of numerator and denominator.
- Exponential functions like \( e^x \) do not have horizontal asymptotes in both directions (e.g., \( e^x \) approaches 0 as \( x \to -\infty \), but diverges as \( x \to \infty \)).
- Trigonometric functions like \( \sin x \), \( \cos x \) do not have horizontal asymptotes because they oscillate indefinitely.
Rules for Rational Functions:
- If degree of numerator < degree of denominator: horizontal asymptote at \( y = 0 \).
- If degree of numerator = degree of denominator: horizontal asymptote at \( y = \frac{\text{leading coeff. of numerator}}{\text{leading coeff. of denominator}} \).
- If degree of numerator > degree of denominator: no horizontal asymptote (may have slant asymptote).
Example:
Find the horizontal asymptote of \( f(x) = \dfrac{3x^2 + 5}{6x^2 – 4x + 7} \).
▶️ Answer/Explanation
Step 1: Compare degrees: both numerator and denominator have degree 2.
Step 2: Ratio of leading coefficients: \( \dfrac{3}{6} = \dfrac{1}{2} \).
Answer: Horizontal asymptote is \( y = \dfrac{1}{2} \).
Example:
Determine \( \lim_{x \to \infty} \dfrac{4x + 1}{x^2 + 2} \).
▶️ Answer/Explanation
Step 1: Degree of denominator (2) is greater than numerator (1).
Step 2: As \( x \to \infty \), fraction approaches 0.
Answer: \( \lim_{x \to \infty} \dfrac{4x + 1}{x^2 + 2} = 0 \). So horizontal asymptote: \( y = 0 \).
Example:
Does the function \( f(x) = e^x \) have a horizontal asymptote? If yes, where?
▶️ Answer/Explanation
Step 1: As \( x \to \infty \), \( e^x \to \infty \) (no asymptote in that direction).
Step 2: As \( x \to -\infty \), \( e^x \to 0 \).
Answer: Horizontal asymptote at \( y = 0 \) as \( x \to -\infty \).
Relative Magnitudes of Functions and Their Rates of Change Using Limits
Limits help us compare the growth rates of two functions as \( x \) becomes very large (or very small). This is especially useful in determining which function grows faster or approaches zero faster.
For two functions \( f(x) \) and \( g(x) \), as \( x \to \infty \), consider:
\( L = \lim_{x \to \infty} \dfrac{f(x)}{g(x)} \).
- If \( L = 0 \), then \( f(x) \) grows slower than \( g(x) \) (or \( g(x) \) dominates).
- If \( L = \infty \), then \( f(x) \) grows faster than \( g(x) \).
- If \( L \) is a finite non-zero number, then \( f(x) \) and \( g(x) \) grow at comparable rates.
Common Growth Rates (Slowest → Fastest):
\( \ln(x) \ll x^a \ll a^x \ll x! \ll x^x \), for \( a > 1 \).
This graph shows exponential growth outpacing linear and polynomial growth. Over the long run, this outpacing will always happen if the base is greater than 1. It may take a long time, however.
Example:
Compare the growth of \( f(x) = \ln x \) and \( g(x) = \sqrt{x} \) as \( x \to \infty \).
▶️ Answer/Explanation
Step 1: Compute \( \lim_{x \to \infty} \dfrac{\ln x}{\sqrt{x}} \).
Step 2: As \( x \to \infty \), \( \ln x \) increases very slowly compared to \( \sqrt{x} \).
\( \lim_{x \to \infty} \dfrac{\ln x}{\sqrt{x}} = 0 \).
Answer: \( \sqrt{x} \) grows faster than \( \ln x \).
Example:
Compare \( f(x) = x^2 \) and \( g(x) = e^x \) as \( x \to \infty \).
▶️ Answer/Explanation
Step 1: Compute \( \lim_{x \to \infty} \dfrac{x^2}{e^x} \).
Step 2: Use the fact that exponential growth dominates polynomial growth.
\( \lim_{x \to \infty} \dfrac{x^2}{e^x} = 0 \).
Answer: \( e^x \) grows faster than \( x^2 \).
Example:
Compare the rates of \( f(x) = e^x \) and \( g(x) = x! \) as \( x \to \infty \).
▶️ Answer/Explanation
Step 1: We know \( x! \) grows much faster than \( e^x \) (factorial dominates exponential).
Step 2: So \( \lim_{x \to \infty} \dfrac{e^x}{x!} = 0 \).
Answer: \( x! \) grows faster than \( e^x \).