Problem 87 - Brief Applied Calculus

Question

True or False: If \(f(2)\) is not defined, then \(\lim _{x \rightarrow 2} f(x)\) does not exist.

Step-by-Step Solution

Answer

False. The limit can exist even if \(f(x)\) is not defined at that point.

Step 1: Understanding the Problem

We need to determine if the statement 'If \(f(2)\) is not defined, then \(\lim _{x \rightarrow 2} f(x)\) does not exist' is true or false. Our goal is to analyze the relationship between function continuity and limits.

Step 2: Analyzing the Limit Concept

Remember, a limit \(\lim_{x \to c} f(x)\) exists if the values of \(f(x)\) approach a particular number as \(x\) approaches \(c\) from both left and right. This limit can exist independent of whether \(f(c)\) is defined.

Step 3: Consider an Example

Consider a function \(f(x)\) which is defined as: \(f(x) = \frac{x^2-4}{x-2}\) for \(x eq 2\). The function \(f(2)\) is not defined, but we can simplify \(f(x)\) to \(f(x) = x+2\) for \(x eq 2\). This simplification helps us to examine the behavior of \(f(x)\) as \(x\) approaches 2.

Step 4: Calculate the Limit

Compute the limit as \(x\) approaches 2: \(\lim_{x \to 2} (x+2) = 4\). Thus, the limit exists even though \(f(2)\) is not defined.

Step 5: Concluding the Solution

From our example, we see that the limit can exist even if the function itself is not defined at that point. Therefore, the original statement is false.

Key Concepts

Understanding Limits

Limits are a foundational concept in calculus. They describe the behavior of a function as its input approaches a particular value. The notation \( \lim_{x \to c} f(x) \) indicates the value that \( f(x) \) nears as \( x \) approaches \( c \). It is crucial to understand that the limit is focused on what happens around \( c \) and not necessarily at \( c \) itself.

To comprehend limits:

• Think of it as observing a vehicle nearing a stop sign; it's about the approach, not necessarily reaching the sign itself.
• Limits can exist even if the function is not defined at that point, as long as the outputs of \( f(x) \) are approaching a specific number from both sides of \( c \).
• Consider the limit as \( x \to 2 \) for \( f(x) = \frac{x^2-4}{x-2} \). Simplifying gives \( f(x) = x+2 \) for \( x eq 2 \), so \( \lim_{x \to 2} f(x) = 4 \), even if \( f(2) \) doesn't exist.

The Concept of Continuity

Continuity in a function means you can draw its graph without lifting your pen. A strict mathematical definition says a function \( f(x) \) is continuous at \( x = c \) if:

• \( f(c) \) is defined.
• The limit \( \lim_{x \to c} f(x) \) exists.
• The limit equals the function's value at that point: \( \lim_{x \to c} f(x) = f(c) \).

But, as we noticed earlier, if \( f(c) \) is not defined, this continuity condition is broken even if the limit exists. For example, if \( f(x) = \frac{x^2-4}{x-2} = x + 2 \) near \( x = 2 \), \( f(2) \) is undefined, so \( f(x) \) isn't continuous at \( x = 2 \), yet the limit is 4.

This example highlights that a function can have an existing limit yet be discontinuous if the function is not defined at that point or the value at that point differs from the limit.

Exploring Undefined Functions

An undefined function means there is no value assigned at a particular point. But, just because a function is undefined at \( x = c \), it doesn't automatically imply limits cannot exist.

Consider situations where specific operations lead to undefined points:

• Division by zero, such as \( \frac{x^2-4}{x-2} \) when \( x=2 \). We simplify this to \( x+2 \) to explore behavior near \( x=2 \), thus sidestepping the undefined spot.
• Inverse functions of a zero angle in trigonometry.

Understanding that the undefined point is not a roadblock to finding limits, consider that you can often simplify, factor, or use substitution methods to examine paths leading to a point. This adaption helps determine limits even amid undefined points.