Calculus Solutions, Important Examples, Formulas and Videos
Algebra of limit
Theorem 1: Let f and g be two functions such that both exist then.
 Limit of sum of two functions is sum of the limits of the function :
 Limit of difference of two functions is difference of the limits of the functions, i.e
 Limit of product of two functions is product of the limits of the functions, i.e
 Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero) , i .e
Note: In particular as case of (iii), when g is the constant function such that g(x) = λ , for some real number λ ,
we have
Limits of polynomials and rational function:
A function f is said to be a polynomial function if f(x) is zero function or if , where are real numbers such that a n ≠ 0 for some natural numbers n.
We know that Hence
Or
A function f is said to be a rational function if are polynomial such that Then ,
Limit of trigonometric function : Let f and g be two real valued functions with the same domain such that f (x) ≤ g(x) for all x in the domain of definition, For some a, if both . in fig 13.8
Theorem:
The following are two important limits
Example: Evaluate
Solution:
Derivatives: uppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by
provided this limit exists. Derivative of f (x)at a is denoted by f ′(a). observe that f′ (a) quantifies the change in f(x) at a with respect to x.
Example: Find the derivative at x = 2 of the function f (x) = 3x.
Solution: The derivative of the function 3x at x = 2 is 3 .
Algebra of derivative of functions: Since the very definition of derivatives involve limits in a rather direct fashion, we expect the rules for derivatives to follow closely that of limits. We collect these in the following theorem .
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