AP Calculus BC 1.14 Connecting Infinite Limits and Vertical Asymptotes Study Notes - New Syllabus
AP Calculus BC 1.14 Connecting Infinite Limits and Vertical Asymptotes Study Notes- New syllabus
AP Calculus BC 1.14 Connecting Infinite Limits and Vertical 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 Infinite Limits and Vertical Asymptotes
Connecting Infinite Limits and Vertical Asymptotes
Connecting Infinite Limits and Vertical Asymptotes
A function \( f(x) \) has a vertical asymptote at \( x = a \) if as \( x \) approaches \( a \) from the left or right, the value of \( f(x) \) increases or decreases without bound (approaches \( +\infty \) or \( -\infty \)).
This is represented as:
\( \lim_{x \to a^-} f(x) = \pm \infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm \infty. \)
Key Idea: If \( \lim_{x \to a^-} f(x) = \pm \infty \) or \( \lim_{x \to a^+} f(x) = \pm \infty \), then the line \( x = a \) is a vertical asymptote of the function.
Common Cases:
- Rational functions where the denominator approaches 0 while numerator stays non-zero.
- Logarithmic functions near their domain boundary (e.g., \( \ln(x) \) as \( x \to 0^+ \)).
Examples of Vertical Asymptotes:
- \( f(x) = \dfrac{1}{x – 2} \) has a vertical asymptote at \( x = 2 \).
- \( g(x) = \ln(x) \) has a vertical asymptote at \( x = 0 \).
Example:
Find the vertical asymptotes of: \( f(x) = \dfrac{x^2 – 1}{x^2 – 4}. \)
▶️ Answer/Explanation
Step 1: Denominator \( x^2 – 4 = (x – 2)(x + 2) \).
Step 2: The function is undefined at \( x = 2 \) and \( x = -2 \).
Step 3: Check limits: \( \lim_{x \to 2^-} f(x) = \pm \infty, \quad \lim_{x \to 2^+} f(x) = \pm \infty. \) Similarly for \( x = -2 \).
Answer: Vertical asymptotes at \( x = 2 \) and \( x = -2 \).
Example:
Determine whether \( f(x) = \ln(x – 3) \) has a vertical asymptote. If yes, where?
▶️ Answer/Explanation
Step 1: Domain of \( \ln(x – 3) \) is \( x > 3 \).
Step 2: As \( x \to 3^+ \), \( \ln(x – 3) \to -\infty \).
Answer: Vertical asymptote at \( x = 3 \).
Example:
Evaluate: \( \lim_{x \to 4^-} \dfrac{1}{(x – 4)^2}. \)
▶️ Answer/Explanation
Step 1: For \( x \) near 4, \( (x – 4)^2 > 0 \).
Step 2: As \( x \to 4^- \), \( (x – 4)^2 \to 0^+ \).
So: \(\dfrac{1}{(x – 4)^2} \to +\infty\).
Answer: Limit is \( +\infty \), vertical asymptote at \( x = 4 \).