Integral Calculus - Limit Of A Function In Ode

Integral Calculus - Limit of a function in ODE

In differential calculus, we studied limits of functions at a point
In integral calculus, we study the limit of a function as its argument approaches a certain value
Limit of a function in ODE (Ordinary Differential Equation) is an important concept in solving ODEs
Let’s explore this concept further in this lecture

Definition

The limit of a function f(x) as x approaches a certain value a is denoted as:

lim (x->a) f(x)
It represents the value that the function approaches as x gets arbitrarily close to a

Evaluating the Limit

There are various techniques to evaluate the limit of a function:
1. Direct Substitution Method
2. Factorization
3. L’Hospital’s Rule
4. Squeeze Theorem
5. Graphical Interpretation
We will discuss each of these techniques in detail

Direct Substitution Method

In this method, we substitute the value for x directly into the function to evaluate the limit
Example:

lim (x->2) (x^2 + x - 6) Substitute x = 2:

= (2^2 + 2 - 6) = (4 + 2 - 6) = 0

Factorization

In this method, we factorize the function and cancel out common terms to simplify the expression
Example:

lim (x->3) (x^2 - 9) / (x - 3) Factorize numerator and cancel out common terms:

= lim (x->3) (x - 3)(x + 3) / (x - 3) = lim (x->3) (x + 3) = 6

L’Hospital’s Rule

L’Hospital’s rule is a technique used to evaluate limits involving indeterminate forms
Indeterminate forms include 0/0, ∞/∞, 0*∞, ∞ - ∞, etc.
The rule states that if the limit of the ratio of two functions f(x)/g(x) is indeterminate, then the limit of their derivatives f’(x)/g’(x) will give the same result
Example:

lim (x->0) sin(x)/x Apply L’Hospital’s rule:

= lim (x->0) cos(x)/1 = cos(0)/1 = 1

Squeeze Theorem

The Squeeze theorem is used to evaluate limits by comparing the function with two other functions whose limits are known
If the function lies between these two functions for all x, then the limit of the function is the same as the limits of the other two functions
Example:

lim (x->0) x*sin(1/x) Apply the Squeeze theorem:

-1 <= sin(1/x) <= 1 -x <= x*sin(1/x) <= x As x approaches 0, the function is squeezed between -x and x:

lim (x->0) -x <= lim (x->0) x*sin(1/x) <= lim (x->0) x = 0 <= lim (x->0) x*sin(1/x) <= 0 Therefore, the limit is 0

Graphical Interpretation

Graphical interpretation of a function can also help to evaluate its limit
The limit of a function at a point corresponds to the y-value of the function as x approaches that point on the graph
Example: In the given graph, the limit of f(x) as x approaches 2 is 3

Summary

The limit of a function in ODE represents the value that the function approaches as its argument approaches a certain value
It can be evaluated using various techniques such as direct substitution, factorization, L’Hospital’s rule, squeeze theorem, and graphical interpretation
These techniques help in solving ODEs and understanding the behavior of functions near certain points

Technique - Direct Substitution Method

Substitute the value for x directly into the function to evaluate the limit
Example:
- lim (x->4) (2x - 5)
  - Substitute x = 4:
    - = (2 * 4 - 5)
    - = (8 - 5)
    - = 3
In this example, the limit of the function as x approaches 4 is 3

Technique - Factorization

Factorize the function and cancel out common terms to simplify the expression
Example:
- lim (x->1) (x^2 - 4) / (x - 1)
  - Factorize numerator and cancel out common terms:
    - = lim (x->1) (x - 2)(x + 2) / (x - 1)
    - = lim (x->1) (x + 2)
    - = 3
In this example, the limit of the function as x approaches 1 is 3

Technique - L’Hospital’s Rule

L’Hospital’s rule is used when the limit involves indeterminate forms (0/0, ∞/∞, 0*∞, ∞ - ∞, etc.)
If the limit of the ratio of two functions f(x)/g(x) is indeterminate, then the limit of their derivatives f’(x)/g’(x) will give the same result
Example:
- lim (x->1) (x^2 - 1) / (x - 1)
  - Apply L’Hospital’s rule by finding the derivatives numerator and denominator:
    - = lim (x->1) 2x / 1
    - = 2
In this example, the limit of the function as x approaches 1 is 2

Technique - Squeeze Theorem

Squeeze theorem is used to evaluate limits by comparing the function with two other functions whose limits are known
If the function lies between these two functions for all x, then the limit of the function is the same as the limits of the other two functions
Example:
- lim (x->0) x^2 * sin(1/x)
  - Apply the Squeeze theorem:
    - -1 <= sin(1/x) <= 1
    - -x^2 <= x^2 * sin(1/x) <= x^2
  - As x approaches 0, the function is squeezed between -x^2 and x^2:
    - lim (x->0) -x^2 <= lim (x->0) x^2 * sin(1/x) <= lim (x->0) x^2
    - = 0 <= lim (x->0) x^2 * sin(1/x) <= 0
In this example, the limit of the function as x approaches 0 is 0

Technique - Graphical Interpretation

Graphical interpretation of a function helps in evaluating its limit
The limit of a function at a point corresponds to the y-value of the function as x approaches that point on the graph
Example:
- - In the given graph, the limit of f(x) as x approaches -2 is 1
  - This can be observed by looking at the y-value of the function as x approaches -2 on the graph

Applications of Limit of a Function in ODE

The concept of the limit of a function is widely used in various applications, including:
- Determining the behavior of a function near a point
- Solving differential equations
- Calculating rates of change and instantaneous rates of change
- Evaluating infinite series and improper integrals

Example - Behavior of a Function

Consider the function f(x) = (x^2 - 1)/(x - 1)
We can analyze the behavior of this function near x = 1 by evaluating its limit as x approaches 1
lim (x->1) (x^2 - 1) / (x - 1)
- Apply direct substitution method:
  - = lim (x->1) (1^2 - 1) / (1 - 1)
  - = 0/0
- Apply L’Hospital’s rule:
  - = lim (x->1) (2x) / 1
  - = 2
The limit of the function as x approaches 1 is 2, indicating that the function has a vertical asymptote at x = 1

Example - Solving a Differential Equation

Consider the differential equation dy/dx = 3x^2 - 2x
To solve this differential equation, we need to determine the function y by integrating the given equation
We can use the initial condition y(0) = 1 to find the specific solution
Evaluate the limit as x approaches 0:
- lim (x->0) (3x^2 - 2x)
- Apply direct substitution method:
  - = lim (x->0) (3*0^2 - 2*0)
  - = 0
The limit of the given function as x approaches 0 does not depend on x, indicating that the solution to the differential equation does not change near x = 0

Example - Calculating Rates of Change

The limit of a function can also be used to calculate rates of change, such as velocity and acceleration
Consider the function s(t) = 5t^2 + 3t + 2, representing the position of an object at time t
To find the velocity of the object at time t, we find the derivative of the position function s(t)
Velocity is the rate of change of position with respect to time
Evaluate the limit as h approaches 0:
- lim (h->0) (s(t + h) - s(t)) / h
- Substitute the given function:
  - = lim (h->0) (5(t + h)^2 + 3(t + h) + 2 - (5t^2 + 3t + 2)) / h
- Simplify the expression:
  - = lim (h->0) (10th + 10h^2 + 3h) / h
  - = lim (h->0) (10t + 10h + 3)
  - = 10t + 3
The velocity of the object at time t is given by the limit as h approaches 0, which is 10t + 3

Example - Evaluating Improper Integrals

The limit of a function can be used to evaluate improper integrals, which have infinite limits of integration or unbounded integrands
Consider the improper integral ∫(0 to ∞) e^(-x) dx
To evaluate this improper integral, we need to find the limit as b approaches ∞ of the definite integral from 0 to b
Evaluate the limit:
- lim (b->∞) ∫(0 to b) e^(-x) dx
- Integrate the given function:
  - = lim (b->∞) [-e^(-x)](0 to b)
  - = lim (b->∞) (-e^(-b) - (-e^0))
  - = lim (b->∞) (-e^(-b) + 1)
  - = 1
The value of the improper integral is 1, indicating that the area under the curve approaches 1 as the upper limit of integration approaches infinity

Real-Life Applications

The concept of the limit of a function has various real-life applications, such as:
- Predicting population growth and decay
- Modeling the spread of diseases
- Analyzing the behavior of financial investments over time
- Understanding the convergence and divergence of infinite series
- Estimating the limit of a physical quantity as it approaches certain conditions

Example - Population Growth

The population of a city is modeled by the function P(t) = 2500 / (1 + 9e^(-0.5t)), where t represents time in years
To determine the population growth rate, we can evaluate the limit as t approaches infinity:
- lim (t->∞) 2500 / (1 + 9e^(-0.5t))
The limit of the population function as t approaches infinity will give us the estimated maximum population of the city

Example - Spread of Diseases

The spread of a contagious disease can be modeled by the function N(t) = N_0 / (1 + be^(-kt)), where t represents time in days
To analyze the spread of the disease, we can evaluate the limit as t approaches infinity:
- lim (t->∞) N_0 / (1 + be^(-kt))
The limit of the disease spread function as t approaches infinity will give us the estimated maximum number of infected individuals

Example - Financial Investments

The value of a financial investment can be modeled by the function V(t) = P(1 + r/n)^(nt), where t represents time in years, P is the principal amount, r is the annual interest rate, and n is the number of times interest is compounded per year.
To analyze the growth of the investment, we can evaluate the limit as t approaches infinity:
- lim (t->∞) P(1 + r/n)^(nt)
The limit of the investment function as t approaches infinity will give us the estimated maximum value of the investment

Example - Convergence of Infinite Series

Infinite series can have different behaviors, such as convergence or divergence
To determine the behavior of an infinite series, we can evaluate the limit of its terms as the number of terms approaches infinity
Example:
- Consider the series 1/2 + 1/4 + 1/8 + …
- To determine if this series converges or diverges, we evaluate the limit of its terms:
  - lim (n->∞) 1/2^n
If the limit of the terms is a finite number, the series converges; otherwise, it diverges

Example - Limit of Physical Quantity

In physics, the concept of limit is often used to determine the behavior of physical quantities as certain conditions are approached
Example:
- Consider a particle moving along a line with position function s(t) = 3t^2 - 2t + 5
- To determine the limit of the particle’s position as time approaches infinity, we evaluate:
  - lim (t->∞) (3t^2 - 2t + 5)
The limit will give us the final position or destination of the particle as time approaches infinity

Limit Laws

The limit laws are important concepts that help in solving more complex limit problems
These laws provide a set of rules for evaluating limits of functions
The basic limit laws include:
- Sum Law: lim (x->a) (f(x) + g(x)) = lim (x->a) f(x) + lim (x->a) g(x)
- Difference Law: lim (x->a) (f(x) - g(x)) = lim (x->a) f(x) - lim (x->a) g(x)
- Constant Multiple Law: lim (x->a) (c * f(x)) = c * lim (x->a) f(x)
- Product Law: lim (x->a) (f(x) * g(x)) = lim (x->a) f(x) * lim (x->a) g(x)
- Quotient Law: lim (x->a) (f(x) / g(x)) = (lim (x->a) f(x)) / (lim (x->a) g(x))

Limit Laws (continued)

Additional limit laws include:
- Power Law: lim (x->a) (f(x))^n = (lim (x->a) f(x))^n
- Exponential Law: lim (x->a) e^(f(x)) = e^(lim (x->a) f(x))
- Logarithmic Law: lim (x->a) log(base b) (f(x)) = log(base b) (lim (x->a) f(x))
- Trigonometric Law: lim (x->a) sin(f(x)) = sin(lim (x->a) f(x))
- Composition Law: lim (x->a) g(f(x)) = g(lim (x->a) f(x)), provided g is continuous at lim (x->a) f(x)

Limit Laws (Examples)

Example:
- Consider the limit lim (x->2) (3x^2 + 4x - 5)
  - Apply the sum law: lim (x->2) 3x^2 + lim (x->2) 4x - lim (x->2) 5
  - Evaluate each term separately: (3 * 2^2) + (4 * 2) - 5
  - Simplify: 12 + 8 - 5 = 15
Example:
- Consider the limit lim (x->3) (x^2 - 9) / (x - 3)
  - Apply the difference law: lim (x->3) (x^2 - 9) - lim (x->3) (x - 3) / (x - 3)
  - Factorize numerator and cancel out common terms: (x - 3)(x + 3) / (x - 3)
  - Simplify: (3 +