Give an example of a function and a point x = a such that the limit of the function as approaches a exists but x = a is not in the domain of the function
Give an example of a function and a point x = a such that the limit of the function as approaches a exists but x = a is not in the domain of the function
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4.4
![**Example of a Function with a Limit at a Point Not in Its Domain**
Consider a function where the limit exists as \( x \) approaches a certain point \( x = a \), but \( a \) is not included in the domain of the function.
One common example is the function \( f(x) = \frac{\sin x}{x} \). Let's look at this function in detail.
**Example:**
1. **Function Definition**: \( f(x) = \frac{\sin x}{x} \)
2. **Domain**: The function is not defined at \( x = 0 \) because attempting to evaluate \( \frac{\sin 0}{0} \) results in an undefined expression (division by zero).
3. **Limit at \( x = 0 \)**:
- As \( x \to 0 \), the limit of \( f(x) = \frac{\sin x}{x} \) is known and proven to be \( 1 \).
- \[
\lim_{x \to 0} \frac{\sin x}{x} = 1
\]
In this case, even though the limit of the function as \( x \) approaches 0 exists and is equal to 1, the function itself is not defined at \( x = 0 \). Thus, \( x = 0 \) is not in the domain of \( f(x) \). This illustrates a scenario where a function has a limit at a point that is not included in its domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8975dae8-fbcb-4b49-8e3a-715e0e53990f%2F04487d44-4072-4605-9169-9d1353201a5a%2Fk5m6zq9_processed.png&w=3840&q=75)
Transcribed Image Text:**Example of a Function with a Limit at a Point Not in Its Domain**
Consider a function where the limit exists as \( x \) approaches a certain point \( x = a \), but \( a \) is not included in the domain of the function.
One common example is the function \( f(x) = \frac{\sin x}{x} \). Let's look at this function in detail.
**Example:**
1. **Function Definition**: \( f(x) = \frac{\sin x}{x} \)
2. **Domain**: The function is not defined at \( x = 0 \) because attempting to evaluate \( \frac{\sin 0}{0} \) results in an undefined expression (division by zero).
3. **Limit at \( x = 0 \)**:
- As \( x \to 0 \), the limit of \( f(x) = \frac{\sin x}{x} \) is known and proven to be \( 1 \).
- \[
\lim_{x \to 0} \frac{\sin x}{x} = 1
\]
In this case, even though the limit of the function as \( x \) approaches 0 exists and is equal to 1, the function itself is not defined at \( x = 0 \). Thus, \( x = 0 \) is not in the domain of \( f(x) \). This illustrates a scenario where a function has a limit at a point that is not included in its domain.
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