Estimating Limit Values from Graphs - AP Calculus BC Study Guide

The Core Idea: Estimating Limit Values from Graphs

The concept of a limit is fundamental to calculus and is used to describe the behavior of a function as its input (the independent variable) gets closer and closer to a particular value. When we analyze the graph of a function, we are not just interested in the value of the function at a specific point, but also in the trend or the value the function is approaching as we get arbitrarily close to that point. Estimating a limit from a graph is a visual exercise in predicting this target $y$-value.

To do this, we must consider the behavior of the function from both sides of the target $x$-value—approaching from the left (values less than our target) and from the right (values greater than our target). A crucial understanding is that the actual value of the function at the point, which could be a filled-in dot, a hole, or even undefined, has no bearing on the existence or value of the limit itself. The limit is concerned only with the journey, not the arrival.

Key Definitions

The concept of a limit is formalized through specific notation that describes the direction of approach.

• The Right-Sided Limit: This is the value that $f(x)$ approaches as $x$ gets closer to $c$ from values greater than $c$. It is denoted as:

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

• The Left-Sided Limit: This is the value that $f(x)$ approaches as $x$ gets closer to $c$ from values less than $c$. It is denoted as:

  $$\underset{x\to c^{-}}{\lim }f(x)=M$$

• The Two-Sided Limit (or simply, The Limit): This is the value that $f(x)$ approaches as $x$ gets closer to $c$ from both sides. It is denoted as:

  $$\underset{x\to c}{\lim }f(x)$$

  The existence of this limit is entirely dependent on the agreement of the left- and right-sided limits.

• Condition for the Existence of a Limit: A two-sided limit exists if and only if the left-sided limit and the right-sided limit both exist and are equal.

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

  If ${\lim }_{x\to c^{-}}f(x)\ne {\lim }_{x\to c^{+}}f(x)$, we say that the limit as $x$ approaches $c$Does Not Exist (DNE).

Understanding the Limit vs. the Function Value

One of the most critical and often misunderstood concepts in limits is the distinction between the value of a limit at a point and the value of the function at that same point. The Essential Knowledge for this topic states explicitly: The existence of a limit at a particular value of $x$ does not depend on the value of the function at $x$.

Let's break this down by considering the three common scenarios you will see on a graph at a point $x=c$:

1. The function is continuous at $x=c$: The graph passes through the point $(c,f(c))$ without any breaks, jumps, or holes. In this case, the value the function approaches from the left is the same as the value it approaches from the right, and this value is also the same as the actual function value, $f(c)$. Here, ${\lim }_{x\to c}f(x)=f(c)$.

2. The function has a hole (removable discontinuity) at $x=c$: The graph approaches the same $y$-value from both the left and the right, but at $x=c$ itself, there is an open circle. This indicates that either $f(c)$ is undefined, or $f(c)$ is defined to be some other value (represented by a solid dot elsewhere on the vertical line $x=c$). In this situation, the limit exists and is equal to the $y$-coordinate of the hole. The value of $f(c)$ is irrelevant to the value of the limit.

3. The function has a jump (jump discontinuity) at $x=c$: The graph approaches one $y$-value from the left and a different $y$-value from the right. Because the left-sided limit does not equal the right-sided limit, the overall two-sided limit Does Not Exist (DNE). This is true regardless of whether $f(c)$ is defined (as a solid dot on one of the two branches) or undefined.

The key takeaway is that finding ${\lim }_{x\to c}f(x)$ is an investigation of the region around$x=c$, not the point $x=c$ itself.

Core Concepts & Rules

• Limits Describe Approaching Behavior: A limit is used to determine the intended $y$-value of a function as the $x$-value gets infinitely close to a specific point.

• Two-Sided Approach is Mandatory: To estimate a limit from a graph, you must trace the function's path as x approaches the target value from the left side and separately from the right side.

• The Limit and Function Value are Independent: The value of ${\lim }_{x\to c}f(x)$ is determined by the $y$-values near $x=c$, not by the value of $f(c)$. The function does not even need to be defined at $c$ for the limit to exist.

• The Existence Condition: The two-sided limit, ${\lim }_{x\to c}f(x)$, exists if and only if the left-sided limit equals the right-sided limit (${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$). If they are not equal, the limit Does Not Exist (DNE).

Step-by-Step Example 1: Basic Application

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