Calculus: Limits and Derivatives

Definition

A limit is the value that a function "approaches" as the input (x) gets closer and closer to some specific number. Crucially, a limit isn't about what happens at that number, but what happens as you get infinitely close to it.

Explanation

Imagine you are walking toward a wall. Each step you take covers half the remaining distance. You will get closer and closer to the wall, infinitely close, even if you never technically touch it. In calculus, that wall is your "limit."

• Existence: A limit exists only if the function approaches the same value from both the left side and the right side.
• Indeterminate Forms: Sometimes, plugging in a number results in 0/0 or ∞∞. This doesn't mean the limit doesn't exist; it just requires further algebraic work (like factoring).

Examples

• Basic Limit: lim_{x → 3} (x + 5). As x gets closer to 3, x + 5 clearly gets closer to 8.
• Factoring Limit: lim_{x → 2} (x² - 4) / (x - 2). If you plug in x=2, you get 0/0. Factoring the top to (x-2)(x+2) allows the (x-2) to cancel, leaving x+2. As x → 2, the value is 4.

Real-Life Use: Terminal Velocity

When a skydiver jumps, air resistance increases with speed. Eventually, air resistance nearly equals gravity. The speed approaches a constant value called "terminal velocity." Mathematically, the limit of the velocity function as time approaches infinity is the terminal velocity.

Algebra of Limits

If lim_{x → a} f(x) = L and lim_{x → a} g(x) = M, then:

• Sum Rule: lim [f(x) + g(x)] = L + M
• Product Rule: lim [f(x) · g(x)] = L · M
• Quotient Rule: lim [f(x) / g(x)] = L / M (where M ≠ 0)

Trigonometric Limits

Memorize these two "golden" limits for exams:

• lim_{x → 0} (sin x / x) = 1
• lim_{x → 0} (1 - cos x) / x = 0

Example

Find lim_{x → 0} [sin(5x) / x]:

To use the rule sin(θ)/θ, we need the denominator to match the angle (5x).
5 · limx → 0 [sin(5x) / 5x]
= 5 · (1)
= 5
        

Definition

A derivative is the "instantaneous rate of change." While an average speed tells you how far you went over an hour, the derivative tells you exactly what your speedometer says at one specific second. It is the slope of the tangent line to a curve at a given point.

Key Points

• Written as dy/dx or f'(x).
• If derivative is positive, the function is increasing.
• If derivative is negative, the function is decreasing.

Real-Life Use: Marginal Cost

In Economics, the derivative of a Total Cost function tells a manager the Marginal Cost—the exact cost increase if they produce just one more unit. This is vital for production decisions.

Rules of Differentiation

Special Functions

• Trigonometric: d/dx(sin x) = cos x | d/dx(cos x) = -sin x | d/dx(tan x) = sec² x
• Exponential/Log: d/dx(e^x) = e^x | d/dx(ln x) = 1/x

Examples

1. Polynomial: f(x) = 4x³ + 2x² - 5
Solution: (4·3)x² + (2·2)x¹ - 0 = 12x² + 4x

2. Product Rule: y = x² sin x
Solution: Let u = x², v = sin x. dy/dx = x²(cos x) + (sin x)(2x) = x² cos x + 2x sin x

Problem 1 (Limit)

Evaluate lim_{x → 4} (x² - 16) / (x - 4)

1. Step 1: Try direct substitution. (4² - 16) / (4 - 4) = 0/0 (Indeterminate).
2. Step 2: Factor the numerator: x² - 16 = (x-4)(x+4).
3. Step 3: Simplify: (x-4)(x+4) / (x-4) = x+4.
4. Step 4: Apply the limit: lim_{x → 4} (x+4) = 8.

Answer: 8

Problem 2 (Derivative)

Find the derivative of y = ln(x) / x

1. Step 1: Use Quotient Rule where u = ln x and v = x.
2. Step 2: du/dx = 1/x and dv/dx = 1.
3. Step 3: Formula: [v(du/dx) - u(dv/dx)] / v².
4. Step 4: Substitute: [x(1/x) - (ln x)(1)] / x² = (1 - ln x) / x².

Answer: (1 - ln x) / x²

• Software: MATLAB and WolframAlpha are used by engineers for instant complex differentiation.
• Machine Learning: AI relies on Gradient Descent, which uses derivatives to minimize error in neural networks.
• Data Science: Python's SymPy library allows for symbolic differentiation in modeling biological growth.

Consider a rocket with height s(t) = -5t² + 50t.

Phase 1 (Derivative): To find velocity at t=3s, take the derivative: v(t) = s'(t) = -10t + 50. At t=3, v(3) = 20 m/s.

Phase 2 (Limit): Engineers use limits to predict temperature and air density as the rocket approaches specific atmospheric boundaries, ensuring smooth transitions in flight software.

1. lim_{x → 0} (sin x / x) is:
   a) 0 | b) 1 | c) ∞ | d) Undefined
2. The derivative of e^2x is:
   a) e^2x | b) 2e^2x | c) 1/2e^2x | d) 2x e^2x-1
3. If f(x) = x⁵, then f'(x) is:
   a) 5x⁴ | b) 5x⁶ | c) x⁴ | d) 5x
4. The derivative of cos x is:
   a) sin x | b) -sin x | c) sec² x | d) cos x
5. lim_{x → 2} (x² + 3) is:
   a) 5 | b) 7 | c) 4 | d) 0
6. The derivative of a constant is:
   a) 1 | b) The constant | c) 0 | d) x
7. The Product Rule for uv is:
   a) u'v' | b) uv' + vu' | c) u'v - uv' | d) u/v'
8. d/dx(ln x) is:
   a) x | b) e^x | c) 1/x | d) 1
9. lim_{x → 0} (1-cos x)/x is:
   a) 1 | b) 0 | c) -1 | d) ∞
10. The slope of a horizontal line is:
    a) 1 | b) Undefined | c) 0 | d) ∞

Answers: 1(b), 2(b), 3(a), 4(b), 5(b), 6(c), 7(b), 8(c), 9(b), 10(c)

1. Q: Evaluate lim_{x → 3} (x² - 9) / (x - 3).
   Ans: 6
2. Q: Find the derivative of y = 3x⁴ - 5x + 2.
   Ans: 12x³ - 5
3. Q: Differentiate y = x · ln x using the product rule.
   Ans: ln x + 1
4. Q: State the limit lim_{x → 0} [sin(3x) / x].
   Ans: 3
5. Q: Find dy/dx if y = tan x.
   Ans: sec² x
6. Q: Explain the difference between a limit and a derivative.
   Ans: A limit describes the value a function approaches; a derivative describes the rate at which that function changes.
7. Q: Calculate the derivative of y = (x+1) / (x-1).
   Ans: -2 / (x-1)²
8. Q: Determine lim_{x → ∞} (1/x).
   Ans: 0
9. Q: What is the derivative of f(x) = e^x + sin x?
   Ans: e^x + cos x
10. Q: Solve lim_{x → 0} (sin x / x²).
    Ans: Undefined/∞

• Limit: The value a function approaches.
• Derivative: The instantaneous rate of change (slope).
• Power Rule: Bring the power down, subtract one.

Mnemonic for Quotient Rule:
"Low d-High minus High d-Low, over Low-Low."
(Denominator · derivative of Numerator - Numerator · derivative of Denominator) / Denominator²

Trig Sign Trick: Remember that "CO" functions (Cos, Cot, Cosec) always result in a negative derivative.