AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes - New Syllabus
AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes- New syllabus
AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Estimating Limit Values from Tables
Key Concepts:
- Estimating Limit Values from Tables
Estimating Limit Values from Tables
Estimating Limit Values from Tables
Limits can be estimated numerically using a table of function values near the point of interest. This is especially useful when a function is too complicated to evaluate analytically.
Key Idea:
- Choose values of \( x \) that approach the target point \( a \) from both the left (smaller than \( a \)) and the right (larger than \( a \)).
- Compute \( f(x) \) at those values and observe what number the function values are approaching.
- If the values from both sides approach the same number, that is the estimated limit.
Important Note: The function value at \( x = a \) does not affect the limit. The limit depends only on values near \( a \).
Example :
Use the table of values to estimate \( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \).
| \( x \) | \( f(x) = \frac{x^2 – 4}{x – 2} \) |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
▶️ Answer/Explanation
As \( x \) gets closer to 2 from the left and right, \( f(x) \) approaches 4.
\( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \approx 4 \).
Example :
Use the table to estimate \( \lim_{x \to 0} \frac{\sin x}{x} \) (in radians).
| \( x \) | \( f(x) = \frac{\sin x}{x} \) |
|---|---|
| -0.1 | 0.9983 |
| -0.01 | 0.99998 |
| -0.001 | 0.9999998 |
| 0.001 | 0.9999998 |
| 0.01 | 0.99998 |
| 0.1 | 0.9983 |
▶️ Answer/Explanation
The function values approach 1 as \( x \to 0 \) from both sides.
\( \lim_{x \to 0} \frac{\sin x}{x} \approx 1 \).
Example :
The function \( f(x) \) has the following values near \( x = 1 \).
Use the table to estimate \( \lim_{x \to 1^-} f(x) \), \( \lim_{x \to 1^+} f(x) \), and determine if \( \lim_{x \to 1} f(x) \) exists.
| \( x \) | \( f(x) \) |
|---|---|
| 0.9 | 4.9 |
| 0.99 | 4.99 |
| 0.999 | 4.999 |
| 1.001 | 8.001 |
| 1.01 | 8.01 |
| 1.1 | 8.1 |
▶️ Answer/Explanation
From the left: As \( x \to 1^- \), the values approach 5, so \( \lim_{x \to 1^-} f(x) = 5 \).
From the right: As \( x \to 1^+ \), the values approach 8, so \( \lim_{x \to 1^+} f(x) = 8 \).
Since LHL ≠ RHL, the two-sided limit does not exist: \( \lim_{x \to 1} f(x) \text{ does not exist} \).