Continuity

• Continuity is one of the fundamental principles in calculus. A function is said to be continuous in the region [a, b] if the limit of the function as x approaches c is equal to f(c) for all values of c from x = a to x = b. The continuity or discontinuity of functions is an important trait, because the derivatives and integral of a function can be calculated only for continuous functions.
  
  

The Derivative

• The derivative of a function is defined using limits. The derivative of a function f(x) is defined to be the limit as h approaches zero of the difference quotient (f(x + h) - f(x) / h). In other words, the derivative is equivalent to the limit of a function's secant line from x = a to x = b as the gap between a and b shrinks to zero. This limit of the secant line is the tangent line at f(x).
  
  

The Integral

• The integral is defined as the limit of an infinite sum known as the Riemann sum. The Riemann sum is a method of approximating the area under a curve by dividing the region into rectangular or trapezoidal regions whose height is the height of the function and whose width is some interval from x1 to x1, also called delta x, or "x. In the same way that the derivative is the limit as the distance between the two points of the secant shrinks to zero, the integral is the limit as the width of the rectangles shrinks to zero. Since there are now an infinite number of rectangles with infinitely small widths that comprise the area, the integral provides an exact calculation of the area under the curve.
  
  

Infinite Series

• The limit also plays a role in calculation of infinite series. An infinite series is the sum of the terms in a sequence that never terminates. Some series, such as geometric series, in which there is a constant ratio between terms, have a finite sum. This sum is calculating using infinite limits. The limit of the infinite geometric series "1/2 + 1/4 + 1/8 + 1/16 ... " is equal to 1. The value of the mathematical constant e is obtained by taking the limit of the infinite series from n = 0 to n = infinity (∞) 1/n!.