Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Tell whether each statement is true or false. If a function \(f\) is defined at \(x=a,\) then \(\lim f(x)\) is always equal to \(f(a)\)

Step-by-Step Solution

Answer

False.

Step 1: Understanding the Limit Concept

The statement talks about the limit of a function \(f(x)\) as \(x\) approaches a certain point \(a\), and whether it is equal to \(f(a)\), the function's value at that point. This involves understanding what a limit and function value mean separately.

Step 2: Consider the Definition of a Limit

The limit \(\lim_{x \to a} f(x) = L\) means that as \(x\) gets closer and closer to \(a\), \(f(x)\) gets closer to \(L\). However, the limit does not require that \(f(a)\) be equal to \(L\), only that \(f(x)\) is close to \(L\) as \(x\) approaches \(a\).

Step 3: Analyze Function Continuity

A function \(f\) is continuous at \(x = a\) if \(\lim_{x \to a} f(x) = f(a)\). The statement assumes continuity, but functions can also be discontinuous. For example, in cases like a removable discontinuity, \(\lim_{x \to a} f(x)\) might exist and be different from \(f(a)\).

Step 4: Determine the Truth of the Statement

Although a function can be defined at \(x = a\), \(\lim_{x \to a} f(x) = f(a)\) is only true if \(f\) is continuous at that point. Hence, the statement is false because it overlooks the possibility of discontinuities where \(\lim_{x \to a} f(x)eq f(a)\).

Key Concepts

Function Continuity

Continuity is one of the foundational concepts in calculus, and it refers to how a function behaves at certain points or over intervals. For a function to be continuous at a particular point, three main conditions must be met:

• The function must be defined at the point, meaning the output exists and is finite, i.e., \( f(a) \) should be defined.
• The limit of the function as it approaches the point should exist.
• The value of the function at that point must equal the limit, i.e., \( \lim_{x \to a} f(x) = f(a) \).

If any of these conditions are not fulfilled, the function is not continuous at that point. Continuity ensures no "gaps" in the graph of a function around the point. For practical purposes, you can think of a continuous function as one that you can draw without lifting your pencil from the paper.

Removable Discontinuity

A removable discontinuity occurs at a point where a function is not continuous but can be made continuous by redefining the function's value at that point. Typically, this type of discontinuity happens when a function has a "hole" due to an undefined value or a defined value that doesn't match the limit.

Here’s how it typically manifests:

• The function is not continuous, but the left-hand and right-hand limits are equal.
• The limit exists, i.e., \( \lim_{x \to a} f(x) = L \), but \( f(a) eq L \) or is undefined.

A practical example could be a function defined as \( f(x) = \frac{x^2 - 1}{x - 1} \). This is undefined at \( x = 1 \) because the denominator becomes zero. However, you can factor and simplify it to \( f(x) = x + 1 \) for \( x eq 1 \), and the limit as \( x \rightarrow 1 \) is \( 2 \). By setting \( f(1) = 2 \), the discontinuity is removed. The "removal" of discontinuity by redefining \( f(a) \) makes the function continuous.

Limit Definition

The limit of a function is a critical concept that deals with the behavior of the function's output as the input approaches a certain value. Specifically, the limit \( \lim_{x \to a} f(x) = L \) means that as \( x \) gets arbitrarily close to \( a \), \( f(x) \) gets arbitrarily close to \( L \).
Understanding limits helps in analyzing how functions behave near points of interest on their domain. It is essential in understanding how to differentiate and integrate functions. Here are key points about limits:

• Limits can exist, even if the function is not defined at that point.
• The notion of getting arbitrarily close is separate from actually reaching a value.
• Limits consider the value approached, not necessarily the value achieved by the function at that point.

Limits are the bedrock for discussing continuity and differentiability. Without grasping limits, we cannot appropriately determine or discuss the continuity of functions.