AP Calculus BC 1.2 Defining Limits and Using Limit Notation Study Notes - New Syllabus
AP Calculus BC 1.2 Defining Limits and Using Limit Notation Study Notes- New syllabus
AP Calculus BC 1.2 Defining Limits and Using Limit Notation Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
Key Concepts:
- Representing Limits Analytically Using Correct Notation
- Interpret Limits Expressed in Analytic Notation
Representing Limits Analytically Using Correct Notation
Representing Limits Analytically Using Correct Notation
In calculus, a limit describes the value a function approaches as its input approaches a certain point. The concept of a limit allows us to rigorously define derivatives, integrals, and continuity.
Notation for Limits:
The limit of a function \( f(x) \) as \( x \) approaches a value \( a \) is written as:
\( \lim_{x \to a} f(x) = L \)
This means: as \( x \) gets arbitrarily close to \( a \), \( f(x) \) gets arbitrarily close to \( L \).
Important Variations of Limit Notation:
- \(\lim_{x \to a^-} f(x)\) : Left-hand limit (approaching from the left of \(a\)).
- \(\lim_{x \to a^+} f(x)\) : Right-hand limit (approaching from the right of \(a\)).
- \(\lim_{x \to \infty} f(x)\) : Limit as \(x\) approaches infinity.
- \(\lim_{x \to -\infty} f(x)\) : Limit as \(x\) approaches negative infinity.
Key Points:
- If both one-sided limits exist and are equal, then the two-sided limit exists.
- If the function approaches different values from the left and right, the two-sided limit does not exist.
Example:
Evaluate \( \lim_{x \to 3} (2x + 5) \).
▶️ Answer/Explanation
Step 1: Since the function \( 2x + 5 \) is continuous, the limit is found by direct substitution.
\( \lim_{x \to 3} (2x + 5) = 2(3) + 5 = 6 + 5 = 11 \)
Answer: \( 11 \)
Example:
Evaluate \( \lim_{x \to 2^-} \frac{x^2 – 4}{x – 2} \) and \( \lim_{x \to 2^+} \frac{x^2 – 4}{x – 2} \).
▶️ Answer/Explanation
Step 1: Factorize numerator:
\( \frac{x^2 – 4}{x – 2} = \frac{(x – 2)(x + 2)}{x – 2} = x + 2 \quad \text{(for } x \neq 2\text{)} \)
Step 2: Compute left-hand limit:
\( \lim_{x \to 2^-} (x + 2) = 2 + 2 = 4 \)
Step 3: Compute right-hand limit:
\( \lim_{x \to 2^+} (x + 2) = 2 + 2 = 4 \)
Since both one-sided limits are equal, the two-sided limit exists and equals 4.
Example:
Find \( \lim_{x \to \infty} \frac{3x^2 + 2}{5x^2 + 4x + 1} \).
▶️ Answer/Explanation
Step 1: Divide numerator and denominator by \( x^2 \) (highest power):
\( \frac{3x^2 + 2}{5x^2 + 4x + 1} = \frac{3 + \frac{2}{x^2}}{5 + \frac{4}{x} + \frac{1}{x^2}} \)
Step 2: As \( x \to \infty \), all fractions with \( x \) vanish:
\( \frac{3}{5} \)
Answer: \( \frac{3}{5} \)
Interpret Limits Expressed in Analytic Notation
Interpret Limits Expressed in Analytic Notation
To interpret a limit written in analytic notation, you need to understand what it says about the behavior of a function near a certain point. A limit does not describe the value of the function at the point (it may not even exist there), but rather what the function approaches as the input gets close to that point.
General Form:
\( \lim_{x \to a} f(x) = L \)
This means that as \( x \) gets arbitrarily close to \( a \), the values of \( f(x) \) get arbitrarily close to \( L \).
Interpretation Points:
- \( \lim_{x \to a^-} f(x) = L \) : From the left of \( a \), \( f(x) \) approaches \( L \).
- \( \lim_{x \to a^+} f(x) = L \) : From the right of \( a \), \( f(x) \) approaches \( L \).
- \( \lim_{x \to \infty} f(x) = L \) : As \( x \) grows without bound, \( f(x) \) approaches \( L \).
- If \( f(a) \neq L \), the limit still exists. Limit describes behavior, not necessarily the actual value at \( a \).
Example :
Interpret the meaning of \( \lim_{x \to 2} f(x) = 5 \)
▶️ Answer/Explanation
This means: as \( x \) gets arbitrarily close to 2 (from both sides), the function \( f(x) \) gets arbitrarily close to 5. It does not necessarily mean that \( f(2) = 5 \); the function might be undefined at \( x = 2 \) or could have a different value there.
Example :
Interpret \( \lim_{x \to 4^-} f(x) = 10 \)
▶️ Answer/Explanation
This means: as \( x \) approaches 4 from the left (values less than 4), the function \( f(x) \) approaches 10. This says nothing about what happens to the right of 4 or the actual value of \( f(4) \).
Example :
Interpret \( \lim_{x \to \infty} f(x) = 7 \)
▶️ Answer/Explanation
This means: as \( x \) grows without bound (becomes arbitrarily large), the values of \( f(x) \) get arbitrarily close to 7. The function levels off at a horizontal asymptote \( y = 7 \).