Problem 86 - Calculus

Question

In Problems \(84-86,\) explain what is wrong with the statement. If \(\lim _{x \rightarrow c} f(x)\) exists, then \(f(x)\) is continuous at \(x=c\)

Step-by-Step Solution

Answer

The statement overlooks that continuity requires \( f(c) \) to equal \( \lim_{x \to c} f(x) \).

Step 1: Understand Limit and Continuity

A function \( f(x) \) is continuous at a point \( x = c \) if three conditions are met: 1) \( f(c) \) is defined, 2) \( \lim_{x \to c} f(x) \) exists, and 3) \( \lim_{x \to c} f(x) = f(c) \). Merely having a limit at \( c \) does not ensure continuity unless \( f(c) \) itself equals this limit.

Step 2: Identify Missing Condition

The statement ignores the need for \( f(c) \) to be defined and equal to \( \lim_{x \to c} f(x) \). Without verifying that \( f(c) \) equals the limit, continuity at \( x = c \) is not guaranteed. There could exist a case where the limit exists, but \( f(c) \) is either undefined or not equal to this limit.

Step 3: Provide a Counter-Example

Consider the piecewise function: \( f(x) = \begin{cases} x^2, & \text{if } x eq 2 \ 5, & \text{if } x = 2 \end{cases} \). Here, \( \lim_{x \to 2} f(x) = 4 \) because as \( x \) approaches 2, \( f(x) \) approaches \( x^2 \). However, \( f(2) = 5 \), which does not equal the limit. Thus, \( f(x) \) is not continuous at \( x = 2 \).

Key Concepts

Understanding Limits

Limits play a crucial role in calculus and are the foundational concept for continuity. The limit of a function at a point, written as \( \lim_{x \to c} f(x) \), represents the value that \( f(x) \) approaches as \( x \) gets closer and closer to \( c \) from both sides. This is not about what happens exactly at \( c \) itself, but rather the behavior of \( f(x) \) around it.

• The limit is about approaching a value, not necessarily reaching it.
• If the left-hand limit \( \lim_{x \to c^-} f(x) \) and the right-hand limit \( \lim_{x \to c^+} f(x) \) both equal the same value, then the two-sided limit \( \lim_{x \to c} f(x) \) exists.

However, just because the limit exists, it does not guarantee continuity at that point. Other criteria need to be met. Understanding these nuances helps clarify why a limit by itself doesn't ensure continuity.

Exploring Piecewise Functions

Piecewise functions are defined by different expressions based on the value of \( x \). They can be tricky when discussing continuity because the function's rule changes at certain points.

• For a piecewise function, each individual piece can behave differently.
• Continuity needs to be checked at the points where the pieces change.

In the example provided, the function \( f(x) \) is defined as \( f(x) = x^2 \) for all \( x \) except 2, and \( f(x) = 5 \) at \( x = 2 \). The problem arises at \( x = 2 \) because \( f(x) \) switches from \( x^2 \) to 5. Although the limit as \( x \) approaches 2 exists and equals 4, the function actually jumps to 5 at this point, which breaks the continuity.

Counter-Example to Clarify Continuity

Counter-examples are powerful tools for understanding concepts. In our context, they help us see why an assumption might be incorrect.

• Consider the example function \( f(x) = \begin{cases} x^2, & \text{if } x eq 2 \ 5, & \text{if } x = 2 \end{cases} \).
• While \( \lim_{x \to 2} f(x) \) equals 4, due to approaching 2 using the \( x^2 \) rule, \( f(2) = 5 \) breaks the continuity condition of \( f(2) = \lim_{x \to 2} f(x) \).

This highlights that having an existent limit is only one part of being continuous at a point. The function must also belong to the limit at that point. Without all conditions being met, such as the definition of the function itself aligning with the limit, continuity doesn't exist. Counter-examples like this expose the critical importance of every requirement in mathematical definitions.