Problem 24
Question
since erf \((0)=0\) and \(\lim _{x \rightarrow \infty}\) erf \((x)=1,\) we have $$\lim _{t \rightarrow \infty} u(x, t)=50[\text { erf }(0)-\text { erf }(0)]=0$$ and $$\lim _{x \rightarrow \infty} u(x, t)=50[\text { erf }(\infty)-\operatorname{erf}(\infty)]=50[1-1]=0.$$
Step-by-Step Solution
Verified Answer
Both limits evaluate to 0.
1Step 1: Understanding erf Function Properties
The error function, denoted as \( ext{erf}(x) \), has two key properties in this context: \( ext{erf}(0) = 0 \) and \( \lim_{x \to \infty} ext{erf}(x) = 1 \). These properties help us explore the limits of the function \( u(x, t) \).
2Step 2: Analyzing the Limit as t Approaches Infinity
Using the expression \( u(x, t) = 50[\text{erf}(0) - \text{erf}(0)] \), when \( t \to \infty \), the error function values do not change. Therefore, this limit simplifies to \( 50[0 - 0] = 0 \).
3Step 3: Analyzing the Limit as x Approaches Infinity
Consider \( \lim_{x \to \infty} u(x, t) = 50[\text{erf}(\infty) - \text{erf}(\infty)] \). As \( x \to \infty \), both \( \text{erf}(\infty) = 1 \). Thus, the expression becomes \( 50[1 - 1] = 0 \).
Key Concepts
Limits and Their Role in CalculusUnderstanding Infinity in MathematicsProperties of the Error Function
Limits and Their Role in Calculus
Limits describe the behavior of a function as its input approaches a particular value. This foundation is crucial for understanding the concept of continuity, derivatives, and integrals.
A limit aims to predict the output value of a function for an input that gets infinitely close to some number, but doesn’t necessarily reach it. For instance, if we need to evaluate \( \lim_{x \to \infty} f(x) \), we check how \( f(x) \) behaves as \( x \) grows without bound.
This concept can be slightly tricky since limits can sometimes appear not to exist, like when a function approaches different values from each direction. Therefore, understanding limits is essential for tackling more complex topics in calculus and analyzing functions, such as the error function seen in the exercise.
- Limits are fundamental for defining derivatives and integrals.
- They help in understanding the continuity of functions.
- Use limits to infer the behavior of functions as parameters approach specific values or infinity.
A limit aims to predict the output value of a function for an input that gets infinitely close to some number, but doesn’t necessarily reach it. For instance, if we need to evaluate \( \lim_{x \to \infty} f(x) \), we check how \( f(x) \) behaves as \( x \) grows without bound.
This concept can be slightly tricky since limits can sometimes appear not to exist, like when a function approaches different values from each direction. Therefore, understanding limits is essential for tackling more complex topics in calculus and analyzing functions, such as the error function seen in the exercise.
- Limits are fundamental for defining derivatives and integrals.
- They help in understanding the continuity of functions.
- Use limits to infer the behavior of functions as parameters approach specific values or infinity.
Understanding Infinity in Mathematics
Infinity is an abstract concept that represents an unbounded quantity. It is not treated as a number but rather as an idea of something that goes beyond any finite measure.
When working with limits, infinity often comes into play when considering the behavior at the 'edges' of the real number line. For example, when we say \( \lim_{x \to \infty} \text{erf}(x) = 1 \), it suggests that as \( x \) increases without end, the error function stabilizes near 1.
In analyzing functions and equations, infinity allows mathematicians to explore asymptotic behavior. It is a way of describing how functions behave at extreme values or how they can infinitely approach something without ever getting there.
- Infinity helps study the end behavior of functions.
- In calculus, we consider how functions act as variables grow indefinitely.
- Infinity often helps in understanding the range and asymptotes of functions.
When working with limits, infinity often comes into play when considering the behavior at the 'edges' of the real number line. For example, when we say \( \lim_{x \to \infty} \text{erf}(x) = 1 \), it suggests that as \( x \) increases without end, the error function stabilizes near 1.
In analyzing functions and equations, infinity allows mathematicians to explore asymptotic behavior. It is a way of describing how functions behave at extreme values or how they can infinitely approach something without ever getting there.
- Infinity helps study the end behavior of functions.
- In calculus, we consider how functions act as variables grow indefinitely.
- Infinity often helps in understanding the range and asymptotes of functions.
Properties of the Error Function
The error function, denoted as \( \text{erf}(x) \), is a special mathematical function pivotal in probability, statistics, and partial differential equations. It is defined as
\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \; dt \]
A few properties make the error function particularly important in certain mathematical contexts.
Most notably:
- \( \text{erf}(0) = 0 \): This indicates that when \( x \) is zero, the function evaluates to zero.
- \( \lim_{x \to \infty} \text{erf}(x) = 1 \): This implies that as \( x \) grows very large, \( \text{erf}(x) \) approaches 1.
- Symmetry: \( \text{erf}(-x) = -\text{erf}(x) \): This means the error function is an odd function, exhibiting symmetry about the origin.
These properties simplify many integrals and probabilities based on the normal distribution, providing a simplified approach to complex mathematical problems.
\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \; dt \]
A few properties make the error function particularly important in certain mathematical contexts.
Most notably:
- \( \text{erf}(0) = 0 \): This indicates that when \( x \) is zero, the function evaluates to zero.
- \( \lim_{x \to \infty} \text{erf}(x) = 1 \): This implies that as \( x \) grows very large, \( \text{erf}(x) \) approaches 1.
- Symmetry: \( \text{erf}(-x) = -\text{erf}(x) \): This means the error function is an odd function, exhibiting symmetry about the origin.
These properties simplify many integrals and probabilities based on the normal distribution, providing a simplified approach to complex mathematical problems.
Other exercises in this chapter
Problem 22
From the Table of Laplace transforms we have $$\int_{0}^{\infty} e^{-s t} \frac{\sin a t}{t} d t=\arctan \frac{a}{s}$$ and $$\int_{0}^{\infty} e^{-s t} \frac{\s
View solution Problem 23
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View solution Problem 26
(a) We use $$U(x, s)=c_{1} e^{-(s / a) x}+c_{2} e^{(s / a) x}+\frac{v_{0}^{2} F_{0}}{\left(a^{2}-v_{0}^{2}\right) s^{2}} e^{-\left(s / v_{0}\right) x}$$ The con
View solution Problem 27
We use $$U(x, s)=c_{1} e^{-\sqrt{s+h} x}+c_{2} e^{\sqrt{s+h} x}$$ The condition \(\lim _{x \rightarrow \infty} u(x, t)=0\) implies \(\lim _{x \rightarrow \infty
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