AP Calculus BC 1.5 Determining Limits Using Algebraic Properties of Limits Study Notes - New Syllabus
AP Calculus BC 1.5 Determining Limits Using Algebraic Properties of Limits Study Notes- New syllabus
AP Calculus BC 1.5 Determining Limits Using Algebraic Properties of Limits Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Determining Limits Using Algebraic Properties of Limits
Key Concepts:
- Determining Limits Using Algebraic Properties of Limits
Determining Limits Using Algebraic Properties of Limits
Determining Limits Using Algebraic Properties of Limits
The value of a limit can often be determined using the limit laws (algebraic properties). These properties allow us to break down complex expressions into simpler components.
Key Limit Laws:
| Property | Description |
|---|---|
| \( \lim_{x \to a} [f(x) + g(x)] = L + M \) | Sum of limits = Limit of sum |
| \( \lim_{x \to a} [f(x) – g(x)] = L – M \) | Difference of limits = Limit of difference |
| \( \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M \) | Product of limits = Limit of product |
| \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, M \neq 0 \) | Quotient of limits = Limit of quotient |
| \( \lim_{x \to a} c \cdot f(x) = c \cdot L \) | Constant multiple law |
| \( \lim_{x \to a} [f(g(x))] = f\big(\lim_{x \to a} g(x)\big) \) | Limit of a composite function (if continuous) |
Conditions:
- These laws apply when individual limits exist and, in the quotient law, the denominator’s limit is not zero.
- Composite function law applies when the outer function is continuous at the inner limit.
Example :
Compute \( \lim_{x \to 3} (x^2 + 5x – 7) \).
▶️ Answer/Explanation
Apply sum and constant laws:
\( \lim_{x \to 3} x^2 = 3^2 = 9 \)
\( \lim_{x \to 3} 5x = 5(3) = 15 \)
\( \lim_{x \to 3} (-7) = -7 \)
Combine: \( 9 + 15 – 7 = 17 \)
Answer: \( 17 \)
Example :
Find \( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \).
▶️ Answer/Explanation
Direct substitution gives \( \frac{4 – 4}{2 – 2} = \frac{0}{0} \), an indeterminate form.
Factor numerator: \( x^2 – 4 = (x – 2)(x + 2) \).
Simplify: \( \frac{(x – 2)(x + 2)}{x – 2} = x + 2 \) for \( x \neq 2 \).
Now take limit: \( \lim_{x \to 2} (x + 2) = 4 \).
Answer: \( 4 \)
Example :
Evaluate \( \lim_{x \to 0} (\sin x + x^2) \).
▶️ Answer/Explanation
Use sum law:
\( \lim_{x \to 0} \sin x = 0 \) and \( \lim_{x \to 0} x^2 = 0 \).
So, \( \lim_{x \to 0} (\sin x + x^2) = 0 + 0 = 0 \).
Example :
Compute \( \lim_{x \to -2} [7x^3] \).
▶️ Answer/Explanation
Apply constant multiple law: \( 7 \cdot \lim_{x \to -2} (x^3) \).
\( x^3 \) at \( x = -2 \) is \( (-2)^3 = -8 \).
So, \( 7 \cdot (-8) = -56 \).
Answer: \( -56 \).
Example:
Find \( \lim_{x \to 1} \frac{x^2 + 3x – 4}{x – 1} \).
▶️ Answer/Explanation
Direct substitution gives \( \frac{(1)^2 + 3(1) – 4}{1 – 1} = \frac{0}{0} \), so factor numerator.
\( x^2 + 3x – 4 = (x – 1)(x + 4) \).
Simplify: \( \frac{(x – 1)(x + 4)}{x – 1} = x + 4 \) for \( x \neq 1 \).
Take the limit: \( \lim_{x \to 1} (x + 4) = 5 \).
Answer: \( 5 \).
Example:
Evaluate \( \lim_{x \to 0^+} \sqrt{9 + 4x} \).
▶️ Answer/Explanation
The outer function \( \sqrt{\phantom{x}} \) is continuous, so we can substitute the inner limit.
\( \lim_{x \to 0^+} (9 + 4x) = 9 \).
Then, \( \sqrt{9} = 3 \).
Answer: \( 3 \).
Example :
Compute \( \lim_{x \to 2} \frac{5x^2 – 3}{x + 1} \).
▶️ Answer/Explanation
Apply quotient law:
\( \lim_{x \to 2} (5x^2 – 3) = 5(2^2) – 3 = 20 – 3 = 17 \).
\( \lim_{x \to 2} (x + 1) = 2 + 1 = 3 \).
So, \( \frac{17}{3} \).
Answer: \( \frac{17}{3} \approx 5.67 \).
More Workout Examples
Example 1
- Find all vertical asymptotes of the graph of each function.
(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)
(b) \(f\left ( x \right )=\frac{x^{2}-4x-5}{x^{2}-x-2}\)
▶️Answer/Explanation
Solution
(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)
\(\frac{x}{\left ( x+1 \right )\left ( x-1 \right )}\)
Factor the denominator.
The denominator is 0 at x = −1 and x = 1 . The numerator is not 0 at these two points. There are two vertical asymptotes, x = −1 and x = 1 .
(b)
At all values other than x = −1, the graph of f coincides with the graph of \(y=\left ( x-5 \right )/\left ( x-2 \right ).\) So, x = 2 is the only vertical asymptote. At x = −1 the graph is not continuous.
Example 2
- Find the limit.
(a) \(\underset{x\rightarrow \infty }{lim}\frac{x^{3}-4x^{2}+7}{2x^{3}-3x-5}\)
(b) \(\underset{x\rightarrow \infty }{lim}\frac{\sqrt{4x^{2}+6x}}{3x-2}\)
▶️Answer/Explanation
Solution
Example 3
- Find the horizontal asymptotes of the graph of the function \(f\left ( x \right )=\frac{\sqrt[3]{2x^{3}-9}}{x}\).
▶️Answer/Explanation
Solution
Exercises – Limits and Asymptotes
Multiple Choice Questions
Question
- \(\underset{x\rightarrow \infty }{lim}\frac{3+2x^{2}-x^{4}}{3x^{4}-5}=\)
(A) -2 (B) \(-\frac{1}{3}\) (C) \(\frac{1}{5}\) (D) 1
▶️Answer/Explanation
Ans:
1. B
Question
- What is \(\underset{x\rightarrow \infty }{lim}\frac{x^{3}+x-8}{2x^{3}+3x-1}=\)
(A) \(-\frac{1}{2}\) (B) 0 (C) \(\frac{1}{2}\) (D) 2
▶️Answer/Explanation
Ans:
2. C
Question
- Which of the following lines is an asymptote of the graph of \(f\left ( x \right )=\frac{x^{2}+5x+6}{x^{2}-x-12}?\)
I. x = −3
II. x = 4
III. y = 1
(A) II only (B) III only (C) II and III only (D) I, II, and III
▶️Answer/Explanation
Ans:
3. C
Question
- If the horizontal line y = 1 is an asymptote for the graph of the function f , which of the following statements must be true?
(A) \(\underset{x\rightarrow \infty }{lim}f\left ( x \right )=1\)
(B) \(\underset{x\rightarrow 1}{lim}f\left ( x \right )=\infty\)
(C) f (1) is undefined
(D) f (x) = 1 for all x
▶️Answer/Explanation
Ans:
4. A
Question
- If x = 1 is the vertical asymptote and y = −3 is the horizontal asymptote for the graph of the function f , which of the following could be the equation of the curve?
(A) \(f\left ( x \right )=\frac{-3x^{2}}{x-1}\)
(B) \(f\left ( x \right )=\frac{-3\left ( x-1 \right )}{x+3}\)
(C) \(f\left ( x \right )=\frac{-3\left ( x^{2}-1 \right )}{x-1}\)
(D) \(f\left ( x \right )=\frac{-3\left ( x^{2}-1 \right )}{\left (x-1 \right )^{2}}\)
▶️Answer/Explanation
Ans:
5. D
Question
- What are all horizontal asymptotes of the graph of \(y=\frac{6+3e^{x}}{3-3e^{x}}\) in the xy – plane?
(A) y = −1 only
(B) y = 2 only
(C) y = −1 and y = 2
(D) y = 0 and y = 2
▶️Answer/Explanation
Ans:
6. C
Question
- Let \(f\left ( x \right )=\frac{3x-1}{x^{3}-8}.\)
(A) Find the vertical asymptote(s) of f . Show the work that leads to your answer.
(B) Find the horizontal asymptote(s) of f . Show the work that leads to your answer.
▶️Answer/Explanation
Ans:
7. (a) x = 2
(b) y = 0
Question
- Let \(f\left ( x \right )=\frac{sin x}{x^{2}+2x}.\)
(A) Find the vertical asymptote(s) of f . Show the work that leads to your answer.
(B) Find the horizontal asymptote(s) of f . Show the work that leads to your answer.
▶️Answer/Explanation
Ans:
8. (a) x = −2
(b) y = 0