Limits and Derivatives Lesson - Definition, Properties, Formula, and Examples

Lesson Overview

• What are Limits and Derivatives?
• Relationship Between Limits and Derivatives
• Limits of a Function
• Limit Representation
• Limit Formula
• Properties of Limits
• Derivatives of a Function
• Derivative Representation
• Properties of Derivatives
• Derivatives Formula
• Limits and Derivatives with Examples

What are Limits and Derivatives?

In calculus, limits and derivatives are essential concepts used to analyze functions.

1. Limit: Describes the behavior of a function as the input approaches a specific value.
   lim (x → c) f(x) = L
   • Example: lim (x → 2) x² = 4
2. Derivative: Measures the rate of change of a function, representing the slope of the tangent line.
   f'(x) = lim (Δx → 0) [f(x + Δx) - f(x)] / Δx
   • Example: For f(x) = x², f'(x) = 2x

Example 1: Limits

Consider the function f(x) = x²:

lim (x → 3) x² = 9

Graphically, as x approaches 3, f(x) approaches 9.

Example 2: Derivatives

For the function f(x) = x², the derivative is:

f'(x) = 2x

At x = 3:

f'(3) = 6

The graph of f(x) = x² shows that the slope of the tangent at x = 3 is 6.

Fig: Graph of f(x) = x² showing the limit and derivative at x = 3)

Relationship Between Limits and Derivatives

Aspect	Limit	Derivative
Definition	Describes the value a function approaches as the input gets closer to a certain point.	Describes the rate of change of a function at a specific point.
Formula	lim (x → c) f(x) = L	f'(x) = lim (Δx → 0) [f(x + Δx) - f(x)] / Δx
Purpose	To find the behavior of a function near a point.	To measure how quickly a function changes at a point.
Use in Calculus	Used to understand the value a function approaches.	Used to calculate the slope of the tangent to the function at a point.
Example	lim (x → 2) (x²) = 4	f'(x) = 2x (for f(x) = x²)
Graphical Representation	The value the function approaches as x approaches a specific value.	The slope of the tangent line at a given point.
Conceptual Link	Derivatives are calculated using limits.	The derivative is defined as the limit of the average rate of change over a small interval.

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Limits of a Function

The limit of a function describes its behavior as the input approaches a specific value.

The general formula for a limit is:

lim (x → c) f(x) = L

Where:

• f(x) is the function.
• c is the value that x is approaching.
• L is the value that the function approaches.

Limit Representation

We write limits using the symbol:

lim
For example:
lim (x → 2) f(x) = 5
This means that as x gets closer to 2, the function f(x) gets closer to 5.

Limit Formula

The limit formula for a function f(x) as x approaches a value c is written as:

lim (x → c) f(x) = L

Where:

• lim denotes the limit.
• x → c indicates that x is approaching the value c.
• f(x) is the function.
• L is the value that the function approaches.

Important Limit Formulas

1. Basic Polynomial Functions:

2. Sum of Functions:

3. Product of Functions:

4. Quotient of Functions:

5. Power Rule:

Properties of Limits

The properties of limits describe how limits behave in different situations. These properties can be used to simplify calculations and evaluate complex limits.

Limit of a Sum	lim (x → c) [f(x) + g(x)] = lim (x → c) f(x) + lim (x → c) g(x)	The limit of a sum is the sum of the limits.
Limit of a Difference	lim (x → c) [f(x) - g(x)] = lim (x → c) f(x) - lim (x → c) g(x)	The limit of a difference is the difference of the limits.
Limit of a Product	lim (x → c) [f(x) * g(x)] = lim (x → c) f(x) * lim (x → c) g(x)	The limit of a product is the product of the limits.
Limit of a Quotient	lim (x → c) [f(x) / g(x)] = lim (x → c) f(x) / lim (x → c) g(x), if lim (x → c) g(x) ≠ 0	The limit of a quotient is the quotient of the limits, provided the denominator is not zero.
Limit of a Constant	lim (x → c) k = k	The limit of a constant is the constant itself.
Limit of a Power	lim (x → c) [f(x)]ⁿ = [lim (x → c) f(x)]ⁿ	The limit of a power is the power of the limit.
Limit of a Root	lim (x → c) √f(x) = √(lim (x → c) f(x)), if lim (x → c) f(x) ≥ 0	The limit of a root is the root of the limit, provided the value inside the root is non-negative.
Limit of a Product with a Constant	lim (x → c) [k * f(x)] = k * lim (x → c) f(x)	Multiplying a function by a constant does not affect the limit, except scaling the result by the constant.
Squeeze Theorem	If f(x) ≤ g(x) ≤ h(x) for all x near c, and lim (x → c) f(x) = lim (x → c) h(x) = L, then lim (x → c) g(x) = L	If a function is "squeezed" between two others with the same limit, then its limit will be the same as the other two.

Derivatives of a Function

The derivative of a function f(x) is defined as:

f'(x) = lim (h → 0) [f(x + h) - f(x)] / h

Where:

• f'(x) is the derivative of f(x).
• h is a small change in x.
• f(x + h) and f(x) represent the function values at x + h and x.

Derivative Representation

• Leibniz Notation:
  dy/dx = lim (h → 0) [f(x + h) - f(x)] / h
  • dy/dx means the derivative of y with respect to x.
• Lagrange Notation:
  f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
  • f'(x) represents the derivative of the function f(x).
• Higher-Order Derivatives:
  • Second derivative: d²y/dx² or f''(x)
  • Third derivative: d³y/dx³ or f'''(x)
  • n-th derivative: dⁿy/dxⁿ or fⁿ(x)
• Newton's Dot Notation (Used in Physics):
  ẏ = dy/dt (for time-dependent functions) ÿ = d²y/dt² (second derivative with respect to time)

Geometric Representation

The derivative f'(x) represents:

• The slope of the tangent to the curve y = f(x) at a point x.
• The rate of change of the function f(x) as x varies.

Example: Tangent Slope

For f(x) = x², at x = 3:

• f'(x) = 2x
• f'(3) = 2(3) = 6
  The slope of the tangent line at x = 3 is 6.

Fig: Geometric Representation of Derivative

Graphical Representation

Fig: Graphical Representation of Derivative

The graphs above illustrate the following:

1. For f(x) = x²:
   • The blue curve represents the function f(x) = x².
   • The orange dashed line shows the tangent at x = 3, with a slope of 6 (f'(3) = 6).
   • The red point marks the point of tangency at (3, 9).
2. For f(x) = 3x + 5:
   • The green line represents the function f(x) = 3x + 5, which is linear.
   • The red dashed line overlaps with the function itself because the slope (f'(x) = 3) is constant at all points.

Properties of Derivatives

1. Linearity of Derivatives

2. Product Rule

3. Quotient Rule

4. Power Rule

5. Chain Rule

6. Derivative of a Constant

Derivatives Formula

1. Basic Derivatives

2. Trigonometric Functions

3. Exponential and Logarithmic Functions

4. Sum and Difference Rules

5. Product Rule

6. Quotient Rule

7. Chain Rule

Examples Using Formulas:

Limits and Derivatives with Examples

Example 1: Solve a Limit Problem
Problem: Find the limit of f(x) = 3x + 2 as x → 1.

Solution:
We use the basic formula for limits:
lim (x → c) f(x) = f(c)

Here, f(x) = 3x + 2 and x → 1. Substitute x = 1 into the function:
f(1) = 3(1) + 2 = 5

So,
lim (x → 1) (3x + 2) = 5

Explanation: As x gets closer to 1, the value of f(x) gets closer to 5. That's what a limit means.

Example 2: Solve a Derivative Problem
Problem: Find the derivative of f(x) = x².

Solution:
The formula for a derivative is:
f'(x) = lim (h → 0) [(f(x + h) - f(x)) / h]

Here, f(x) = x². Let's calculate step by step:

1. Substitute f(x + h) and f(x):
   f'(x) = lim (h → 0) [(x + h)² - x²] / h
2. Expand (x + h)²:
   (x + h)² = x² + 2xh + h²

Substitute back:
f'(x) = lim (h → 0) [x² + 2xh + h² - x²] / h

3. Simplify the numerator:
   f'(x) = lim (h → 0) [2xh + h²] / h
4. Factor h out of the numerator:
   f'(x) = lim (h → 0) h(2x + h) / h

Cancel h (since h ≠ 0 for a limit):
f'(x) = lim (h → 0) (2x + h)

5. Evaluate the limit as h → 0:
   f'(x) = 2x + 0 = 2x

Answer: The derivative of f(x) = x² is f'(x) = 2x.

Explanation: The derivative tells us the rate at which x² changes. For example, when x = 3, f'(3) = 2(3) = 6, meaning the function is increasing at a rate of 6.

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