Differential_Calculus

1. Limits

1.1 Definition

1.2 Limit laws

• Sum law: the limit of a sum is the sum of the limits
• Product law: the limit of a product is the product of limits
• Quotient law: the limit of quotient is the quotient of limits
• Power law
• Root law
• Direct substitution property: If f is a polynomial or a rational function and a is in the domain of f, then

\[\lim_{x \to a}f(x) = f(a) \]

Functions with the Direct Substitution Property are called continuous at a

• Replacement law
• Squeeze theorem

1.3 Precise definition of infinite limits

2. Continuity

The order of function f and Lim can be reversed.

3. Intermediate Value Theorem

The interval must be closed, or this IVT does not hold. # 4. Derivative of a function h could be larger or less than 0

5. Sequence

Thus, we can evaluate the limit of a sequence by evaluating the limit of corresponding continuous function

6. Taylor Series