Estimating Limit Values from Graphs - AP Calculus AB Study Guide

The Core Idea: Estimating Limit Values from Graphs

The concept of a limit describes the behavior of a function as the input variable gets closer and closer to a particular value. When we analyze a limit from a graph, we are essentially asking: "What is the intended y-value (or height) of the function at a specific x-value, based on the path of the graph leading towards it?" We investigate this by observing the function's behavior as we approach the target x-value from both the left side and the right side.

Crucially, the existence and value of a limit at a point do not depend on the function's actual value at that point. The function could be undefined (like at a hole in the graph) or have a different value entirely, but the limit can still exist. The limit is concerned only with the trend or the approach, not the final destination. A limit exists if and only if the approach from the left and the approach from the right lead to the exact same y-value.

Key Definitions

This topic is based on conceptual definitions rather than formulas. The notation is key to understanding the concepts.

• Two-Sided Limit: The expression ${\lim }_{x\to c}f(x)=L$ means that the value of $f(x)$ gets arbitrarily close to the number $L$ as $x$ approaches $c$ from both sides.

• Left-Sided Limit: The expression ${\lim }_{x\to c^{-}}f(x)=L$ means that the value of $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $c$ from values less than$c$.

• Right-Sided Limit: The expression ${\lim }_{x\to c^{+}}f(x)=L$ means that the value of $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $c$ from values greater than$c$.

• Existence of a Limit: The two-sided limit ${\lim }_{x\to c}f(x)$ exists and is equal to $L$ if and only if the left-sided and right-sided limits both exist and are equal to $L$.

  $$\underset{x\to c}{\lim }f(x)=L⟺\underset{x\to c^{-}}{\lim }f(x)=L\text{ and }\underset{x\to c^{+}}{\lim }f(x)=L$$

Understanding The Limit vs. The Function Value

One of the most fundamental concepts in limits is that the value of the limit as $x$ approaches $c$ is independent of the value of the function at$c$. The limit describes the behavior near a point, while the function value, $f(c)$, is the actual value at that point.

Consider these three common scenarios on a graph:

1. A Hole (Removable Discontinuity): The graph appears continuous except for a single point that is missing.

   • The left-sided limit and the right-sided limit will approach the same y-value of the hole.

   • Therefore, the two-sided limit exists.

   • The function value $f(c)$ is undefined.

   • In this case, ${\lim }_{x\to c}f(x)\ne f(c)$ because $f(c)$ does not exist.

2. A Jump (Jump Discontinuity): The graph approaches one y-value from the left and a different y-value from the right.

   • The left-sided limit and the right-sided limit exist, but they are not equal.

   • Because the one-sided limits disagree, the two-sided limit does not exist.

   • The function value $f(c)$ will be defined by a solid dot on one of the two pieces of the graph.

3. A Point Displacement: The graph is continuous, but the point at $x=c$ is moved to a different y-value, leaving a hole where it "should" be.

   • The left-sided limit and the right-sided limit will approach the same y-value of the hole.

   • Therefore, the two-sided limit exists.

   • The function value $f(c)$ is defined by the solid, displaced dot.

   • In this case, ${\lim }_{x\to c}f(x)\ne f(c)$ because the limit and the function value are different numbers.

Core Concepts & Rules

• A limit describes the intended y-value of a function as the x-value gets arbitrarily close to a specific point.

• To estimate a limit from a graph, you must examine the function's behavior by tracing the curve from both the left side and the right side of the target x-value.

• The limit from the left, ${\lim }_{x\to c^{-}}f(x)$, and the limit from the right, ${\lim }_{x\to c^{+}}f(x)$, must be equal for the overall (two-sided) limit, ${\lim }_{x\to c}f(x)$, to exist.

• If the left- and right-sided limits are not equal, the two-sided limit does not exist (DNE).

• The actual value of the function at the point, $f(c)$ (represented by a solid dot), has no bearing on the existence or value of the limit as $x$ approaches $c$. The limit is about the approach, not the arrival.

Step-by-Step Example 1: Limit at a Hole

Problem: Use the graph of $f(x)$ below to determine the following values:

a) ${\lim }_{x\to 2^{-}}f(x)$

b) ${\lim }_{x\to 2^{+}}f(x)$

c) ${\lim }_{x\to 2}f(x)$

d) $f(2)$

(A graph showing a line passing through $(0,1)$ and $(4,5)$, but with an open circle (hole) at $(2,3)$ and a solid dot at $(2,1)$)

Solution:

Step 1: Analyze the left-sided limit.

To find ${\lim }_{x\to 2^{-}}f(x)$, we trace the graph of $f(x)$ starting from the left of $x=2$ and moving towards $x=2$. As our x-values get closer to 2 (e.g., 1.9, 1.99, 1.999), the y-values on the graph get closer to 3.

Therefore, ${\lim }_{x\to 2^{-}}f(x)=3$.

Step 2: Analyze the right-sided limit.

To find ${\lim }_{x\to 2^{+}}f(x)$, we trace the graph of $f(x)$ starting from the right of $x=2$ and moving towards $x=2$. As our x-values get closer to 2 (e.g., 2.1, 2.01, 2.001), the y-values on the graph also get closer to 3.

Therefore, ${\lim }_{x\to 2^{+}}f(x)=3$.

Step 3: Compare the one-sided limits to determine the overall limit.

Since the left-sided limit and the right-sided limit are both equal to 3, the two-sided limit exists and is equal to 3.

Therefore, ${\lim }_{x\to 2}f(x)=3$.

Step 4: Identify the function value at the point.

To find $f(2)$, we look for the y-value of the solid dot at $x=2$. The graph shows a solid dot at the point $(2,1)$.

Therefore, $f(2)=1$. Note that this is different from the limit value.

Step-by-Step Example 2: Exam-Style Application at a Jump Discontinuity

Problem: The graph of a function $g(x)$ is shown below. Find the value of each expression, or state that it does not exist.

a) ${\lim }_{x\to -1^{-}}g(x)$

b) ${\lim }_{x\to -1^{+}}g(x)$

c) ${\lim }_{x\to -1}g(x)$

d) $g(-1)$

(A graph showing a curve ending at an open circle at $(-1,4)$. A second curve starts at a solid dot at $(-1,-2)$ and continues to the right.)

Solution:

Step 1: Analyze the left-sided limit.

To find ${\lim }_{x\to -1^{-}}g(x)$, we trace the graph as $x$ approaches -1 from the left side. The path of the graph leads us to the open circle at $y=4$. The intended height from the left is 4.

Therefore, ${\lim }_{x\to -1^{-}}g(x)=4$.

Step 2: Analyze the right-sided limit.

To find ${\lim }_{x\to -1^{+}}g(x)$, we trace the graph as $x$ approaches -1 from the right side. This path leads us to the solid dot at $y=-2$. The intended height from the right is -2.

Therefore, ${\lim }_{x\to -1^{+}}g(x)=-2$.

Step 3: Compare the one-sided limits.

We compare the results from Step 1 and Step 2.

${\lim }_{x\to -1^{-}}g(x)=4$

${\lim }_{x\to -1^{+}}g(x)=-2$

Since $4\ne -2$, the left-sided and right-sided limits are not equal.

Therefore, ${\lim }_{x\to -1}g(x)$ does not exist.

Step 4: Identify the function value.

To find $g(-1)$, we look for the y-coordinate of the solid dot at $x=-1$. The graph has a solid dot at $(-1,-2)$.

Therefore, $g(-1)=-2$.

Using Your Calculator

This topic is primarily conceptual and graphical. You will typically be given a graph to analyze. However, if you are given a function rule, you can use your calculator to investigate a limit by creating a graph or a table of values.

**To estimate \lim_{x \to c} f(x)` using a table:** 1. Enter the function into `Y1` in the `Y=` menu. 2. Go to `TBLSET` (Table Setup). Set `TblStart` to a value very close to $c, and set ΔTbl (delta table) to a very small number, like $0.001$.

3. Go to TABLE. Observe the Y1 values as the x-values get closer to $c$.

4. To check the other side, you can manually enter x-values into the table that are very close to $c$. For example, to check ${\lim }_{x\to 2}f(x)$, you could enter $1.99$, $1.999$, $2.001$, and $2.01$ to see if the y-values approach the same number from both sides.

To estimate a limit from a calculator-generated graph:

1. Enter the function into Y1.

2. Use the ZOOM feature (e.g., $ZStandard$ or $ZDecimal$) to get a good viewing window around $x=c$.

3. Use the TRACE button. Enter x-values very close to $c$ from the left (e.g., $c-0.001$) and from the right (e.g., $c+0.001$) and observe if the y-values are approaching the same number.

AP Exam Quick Hit

Common Question Types

• Analyzing a complex piecewise graph: You will be given a single graph with multiple features (holes, jumps, cusps, endpoints) and asked to evaluate a series of one-sided limits, two-sided limits, and function values at various points. This is the most common format.

• Finding a limit from a graph at a point of continuity: You will be asked to find ${\lim }_{x\to c}f(x)$ where the function is continuous. In this case, the limit is simply the y-value of the point on the graph, $f(c)$. This tests whether you understand the concept even in the simplest case.

Common Mistakes

• Confusing the limit with the function value: A student sees a hole at $(3,5)$ and a solid dot at $(3,1)$ and incorrectly states that ${\lim }_{x\to 3}f(x)=1$. The limit is the intended height ($5$), not the actual value ($1$).

• Assuming the limit does not exist at a hole: A student sees a hole at $(3,5)$ and incorrectly concludes that the limit does not exist because $f(3)$ is undefined. The limit does exist and is equal to 5 because the approach from both sides is consistent.

• Averaging the two sides at a jump: At a jump discontinuity where the left limit is 4 and the right limit is -2, a student might incorrectly try to average them or pick one, rather than concluding that the two-sided limit does not exist.

• Ignoring one side: When asked for a two-sided limit ${\lim }_{x\to c}f(x)$, a student only checks the behavior from the left and provides that value, forgetting that the limit only exists if the right side agrees.