basic calculus reviewer

• Limit Notation


  In symbols the limit is written as: "the limit of f of x as x approaches a is b." where a can be any number or infinity, f(x) is any function of x, and b is any number or does not exist (DNE) if the function goes to infinity.
• One-Sided Limit


  The limit of a function as it approaches a specific value of x either from the left side or the right side. Notation: lim(x->a-) f(x) (from the left) and lim(x->a+) f(x) (from the right)
• Limit Laws


  • Limit Theorem 1: The limit of a constant function is the constant.
  • Theorem 2: If f is a polynomial function, then lim(x->a) f(x) = f(a).
  • Limit Theorem 3: The limit of a sum is the sum of the limits.
  • Limit Theorem 4: The limit of a constant times a function is the constant times the limit of the function.
  • Limit Theorem 5: The limit of a product is the product of the limits.
  • Limit Theorem 6: The limit of a quotient is the quotient of the limits.
  • Limit Theorem 7: The limit of a function raised to an exponent is the limit of the function raised to that exponent.
• Evaluating Limit By Rationalization


  Steps to evaluate limits using rationalization
• Evaluating Limits of Piecewise Functions


  Steps to evaluate limits of piecewise functions
• The Squeeze Theorem (Sandwich Theorem)


  A function's limit can be squeezed or sandwiched between two other functions
• Continuity


  A function is discontinuous at a point x = a if at least one of the three conditions was not satisfied
• Types of Discontinuities


  • Removable discontinuity (a graph has a hole)
  • Jump discontinuity (two ends of a function do not meet)
  • Infinite discontinuity (function has a vertical asymptote)
• Tangent line


  A line that touches a curve at only one point
• Secant line


  A line that intersects two or more points on a curve
• Derivative of a Function


  The derivative of a function f(x) with respect to x is f'(x) defined by the limit (f(x+h)-f(x))/h as h approaches 0