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Definition: limit of a function of two variables if for each ?>0 there exists a small enough ?>0 such that for all points (x,y) in a ? disk around (a,b), except possibly for (a,b) itself, the value of f(x,y) is no more than ? away from L (Figure 14.2.

The list of basic integral formulas is given below: ? 1 dx = x + C. ? a dx = ax+ C. ? xn dx = ((xn+1)/(n+1))+C ; n?1. ? sin x dx = ? cos x + C. ? cos x dx = sin x + C. ? sec2x dx = tan x + C. ? csc2x dx = -cot x + C. ? sec x (tan x) dx = sec x + C.

The two definitions of a derivative are as follows: By the geometrical approach: The slope of the curve for the given function is called the derivative of a function. By physical approach: The instantaneous rate of change of a function concerning the variable at a point is called the derivative of a function.

Limits formula:- Let y = f(x) as a function of x. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained unique number is called the limit of f(x) at x = a.

The limit formula is the representation of the behavior of the function at a specific point and the formula analyzes that function. Limit describes the behavior of some quantity that depends on an independent variable, as that independent variable approaches or comes close to a particular value.

Derivatives Formula. f?(x) = limx?a f(x + h) ? f(x) / h is referred to as the f at x if derivative limx?a f(x + h) - f(x) / h exists on a finite scale. For function f, its derivative is defined as f?(x), assuming the existence of the aforementioned equation.

The limit value typically has two types of values, namely, the Left-hand Limit and the Right-Hand Limit. Thus, Left-Hand Limit: limx?a?f(x)=limh?0f(a?h) Right-Hand Limit: limx?a+f(x)=limh?0f(a+h)