1. Limits, Continuity, and Differentiability

Limits

A limit describes the value that a function "approaches" as the input approaches some value. We write lim (x→a) f(x) = L.

• Left-Hand Limit (LHL): The value f(x) approaches as x approaches 'a' from the left (x < a). Written as lim (x→a^-) f(x).
• Right-Hand Limit (RHL): The value f(x) approaches as x approaches 'a' from the right (x > a). Written as lim (x→a⁺) f(x).

Existence of a Limit: A limit L exists at x=a if and only if the Left-Hand Limit is equal to the Right-Hand Limit.
LHL = RHL = L

Continuity

A function is continuous at a point x=a if its graph is "unbroken" at that point. You can draw it without lifting your pen.

Conditions for Continuity at x=a:

1. The function is defined at x=a (i.e., f(a) exists).
2. The limit of the function as x approaches 'a' exists (i.e., LHL = RHL).
3. The limit equals the function's value (i.e., lim (x→a) f(x) = f(a)).

Differentiability

A function is differentiable at a point x=a if it has a well-defined, non-vertical tangent line at that point. It measures the instantaneous rate of change.

A function is differentiable if the limit of the difference quotient exists:

2. Partial Derivatives

For a function of multiple variables, like f(x, y), a partial derivative is the derivative with respect to one variable, while treating all other variables as constants.

• Partial derivative w.r.t. x:
  ∂f/∂x (or f_x)

  How-to: Differentiate f(x, y) as if 'x' is the only variable and 'y' is a constant (like the number 5).

• Partial derivative w.r.t. y:
  ∂f/∂y (or f_y)

  How-to: Differentiate f(x, y) as if 'y' is the only variable and 'x' is a constant.

Example: Let f(x, y) = x³y² + 2x + 5y

• ∂f/∂x = (3x² * y²) + 2 + 0 = 3x²y² + 2
• ∂f/∂y = (x³ * 2y) + 0 + 5 = 2x³y + 5

3. Indeterminate Forms

These are expressions where the limit cannot be determined by simply substituting the value. They require special techniques (like L'Hospital's Rule) to evaluate.

The 7 Indeterminate Forms:

1. 0 / 0 (e.g., sin(x) / x at x=0)
2. ∞ / ∞ (e.g., x² / e^x at x=∞)
3. 0 · ∞
4. ∞ - ∞
5. 0⁰
6. 1^∞
7. ∞⁰

4. L'Hospital's Rule

This rule is a powerful method for evaluating limits of indeterminate forms 0/0 or ∞/∞.

L'Hospital's Rule:
If lim (x→a) [f(x) / g(x)] results in 0/0 or ∞/∞,
AND if the limit of their derivatives exists,
THEN:

lim (x→a) [f(x) / g(x)] = lim (x→a) [f'(x) / g'(x)]

How to use for other forms:

• Form 0 · ∞: (f(x) * g(x))
  Rewrite as f(x) / (1/g(x)) (becomes 0/0) or g(x) / (1/f(x)) (becomes ∞/∞).
• Forms 0⁰, 1^∞, ∞⁰: (f(x)^g(x)) 1. Let y = f(x)^g(x) 2. Take the natural log: ln(y) = g(x) · ln(f(x)). 3. This is now a (0 · ∞) form. Evaluate lim [ln(y)] using the method above. 4. Let's say lim [ln(y)] = L. 5. The original limit is lim [y] = e^L. (Don't forget this last step!)
• Form ∞ - ∞: (f(x) - g(x))
  Use algebra to combine them into a single fraction (e.g., find a common denominator). This will convert it to 0/0 or ∞/∞.

5. Successive Differentiation

This is the process of differentiating a function multiple times. The 'n'-th derivative is the result of differentiating 'n' times.

• First Derivative: y' or f'(x) or dy/dx
• Second Derivative: y'' or f''(x) or d²y/dx²
• n-th Derivative: y⁽ⁿ⁾ or f⁽ⁿ⁾(x) or dⁿy/dxⁿ

Standard n-th derivatives to memorize:

6. Leibnitz's Rule

This rule provides a formula for the n-th derivative of a product of two functions (u · v). It is similar to the binomial expansion.

Leibnitz's Rule:
If y = u · v, where u and v are functions of x, then the n-th derivative is:

y⁽ⁿ⁾ = (u · v)⁽ⁿ⁾ =
ⁿC₀ · u⁽ⁿ⁾ · v⁽⁰⁾ + ⁿC₁ · u^(n-1) · v⁽¹⁾ + ⁿC₂ · u^(n-2) · v⁽²⁾ + ... + ⁿC_n · u⁽⁰⁾ · v⁽ⁿ⁾

Where ⁿC_r = n! / (r! (n-r)!) is the binomial coefficient.

7. Rolle's Theorem and Mean Value Theorem

Rolle's Theorem

A specific case of the Mean Value Theorem.

Conditions: If a function f(x) is:

1. Continuous on the closed interval [a, b],
2. Differentiable on the open interval (a, b), and
3. f(a) = f(b) (the endpoints are at the same height),

Conclusion: ...then there exists at least one point 'c' in the open interval (a, b) such that f'(c) = 0.

Geometrically: If a smooth curve starts and ends at the same height, it must have at least one point with a horizontal tangent (a "peak" or "valley") somewhere in between.

Mean Value Theorem (Lagrange's MVT)

A generalization of Rolle's Theorem.

Conditions: If a function f(x) is:

1. Continuous on the closed interval [a, b], and
2. Differentiable on the open interval (a, b),

Conclusion: ...then there exists at least one point 'c' in the open interval (a, b) such that:

f'(c) = [ f(b) - f(a) ] / [ b - a ]

Geometrically: The term on the right is the slope of the secant line connecting the endpoints (a, f(a)) and (b, f(b)). The term on the left is the slope of the tangent line at 'c'. The theorem guarantees that there is at least one point 'c' where the tangent line is parallel to the secant line through the endpoints.

8. Euler's Theorem on Homogeneous Functions

Homogeneous Function

A function f(x, y) is a homogeneous function of degree 'n' if, for any constant t:

Example: f(x, y) = x³ + 2x²y + y³.
f(tx, ty) = (tx)³ + 2(tx)²(ty) + (ty)³ = t³x³ + 2t³x²y + t³y³ = t³(x³ + 2x²y + y³) = t³ · f(x, y).
This is a homogeneous function of degree 3.

Euler's Theorem:
If f(x, y) is a homogeneous function of degree 'n', then:

x · (∂f/∂x) + y · (∂f/∂y) = n · f

Corollary (for second-order derivatives)