AP Calculus BC 1.3 Estimating Limit Values from Graphs Study Notes - New Syllabus
AP Calculus BC 1.3 Estimating Limit Values from Graphs Study Notes- New syllabus
AP Calculus BC 1.3 Estimating Limit Values from Graphs Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Estimating Limit Values from Graphs
Key Concepts:
- Estimating Limits from Graphs and Understanding One-Sided Limits
Estimating Limits from Graphs and Understanding One-Sided Limits
Estimating Limits from Graphs and Understanding One-Sided Limits
The graph of a function provides a visual way to estimate limits. From the graph, you can identify how the function behaves as \( x \) approaches a particular value from the left or from the right.
This leads to the concepts of Left-Hand Limit (LHL) and Right-Hand Limit (RHL).
Definitions:
Left-Hand Limit (LHL):
\( \lim_{x \to a^-} f(x) \) is the value the function approaches as \( x \) approaches \( a \) from the left (smaller values of \( x \)).
Right-Hand Limit (RHL):
\( \lim_{x \to a^+} f(x) \) is the value the function approaches as \( x \) approaches \( a \) from the right (larger values of \( x \)).
The two-sided limit \( \lim_{x \to a} f(x) \) exists if and only if the LHL and RHL both exist and are equal.
Limits at Infinity (One-Sided):
- \( \lim_{x \to \infty} f(x) \): Describes the behavior of \( f(x) \) as \( x \) grows very large (to the right on the graph).
- \( \lim_{x \to -\infty} f(x) \): Describes the behavior of \( f(x) \) as \( x \) becomes very negative (to the left on the graph).
- These limits are often used to identify horizontal asymptotes.
Key Observations from Graphs:
- If the graph approaches the same height from both sides near \( x = a \), the limit exists.
- If the left and right behaviors differ, the two-sided limit does not exist, but one-sided limits may exist individually.
- If the graph goes to infinity or oscillates near \( x = a \), the limit does not exist.
Example :
From the graph, as \( x \) approaches 2 from both sides, \( f(x) \) approaches 5.
Estimate \( \lim_{x \to 2} f(x) \), \( \lim_{x \to 2^-} f(x) \), and \( \lim_{x \to 2^+} f(x) \).
▶️ Answer/Explanation
The graph shows that from the left and right, the function approaches 5.
\( \lim_{x \to 2^-} f(x) = 5 \)
\( \lim_{x \to 2^+} f(x) = 5 \)
Since both one-sided limits are equal, the two-sided limit exists: \( \lim_{x \to 2} f(x) = 5 \).
Example:
From the graph, as \( x \) approaches 3 from the left, \( f(x) \) approaches 4. From the right, it approaches 7.
Does \( \lim_{x \to 3} f(x) \) exist?
▶️ Answer/Explanation
\( \lim_{x \to 3^-} f(x) = 4 \) and \( \lim_{x \to 3^+} f(x) = 7 \).
Since they are not equal, the two-sided limit does not exist. However, both one-sided limits exist individually.
Example:
The graph of \( f(x) = \frac{1}{x} \).
Estimate \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \).
▶️ Answer/Explanation
As \( x \to \infty \), \( f(x) = \frac{1}{x} \to 0^+ \).
As \( x \to -\infty \), \( f(x) = \frac{1}{x} \to 0^- \).
So both limits exist, and the horizontal asymptote is \( y = 0 \).
Example:
The graph of the piecewise function is given as:
\(f(x) = \begin{cases} 3x, & x < 0 \\ 12 – 2x, & x \ge 0 \end{cases} \)
Using the graph, find:
- \( \lim_{x \to 0^-} f(x) \)
- \( \lim_{x \to 0^+} f(x) \)
- Does \( \lim_{x \to 0} f(x) \) exist?
▶️ Answer/Explanation
(a)For \( x < 0 \), \( f(x) = 3x \). As \( x \to 0^- \), \( f(x) \to 3(0) = 0 \).
\( \lim_{x \to 0^-} f(x) = 0 \).
(b) For \( x \ge 0 \), \( f(x) = 12 – 2x \). As \( x \to 0^+ \), \( f(x) \to 12 – 2(0) = 12 \).
\( \lim_{x \to 0^+} f(x) = 12 \).
(c) LHL = 0 and RHL = 12. Since they are not equal, \( \lim_{x \to 0} f(x) \) does not exist.