Topic 5. Limiting value of functions PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó Mathematics I. Lectures BSc materials

Limit of functions I. Finite limit at finite value Examine both functions near at x o =1. D f : x≥0 x≠1 Consider and Consider different sequences x n convergent with limit 1: but x n ≠ 1 X 2 =1,5 f(x 3 )=2,15 X 3 =1,33 f(x 2 )=2,25 X 10 =1,1 X 0 =1 f(x 10 )=2,05 X 2 =1,5 g(x 3 )=2,15 X 3 =1,33 g(x 2 )=2,25 X 10 =1,1 X 0 =1 g(x 10 )=2,05 For arbitrary x n with x n → 1 x n ≠ 1 x n ≠ x o X 1 ’=0,5 g(x 1 ’ )=1,7 X 4 ’=0,8 g(x 4 ’ )=1,89 X 10 ’=0,91 g(x 10 ’ )=1,95 X 1 ’=0,5 f(x 1 ’ )=1,7 X 4 ’=0,8 f(x 4 ’ )=1,89 X 10 ’=0,91 f(x 10 ’ )=1,95 f(x) g(x) PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

May be defined separately the right –hand side and the left – hand side limits. Definition: Right-hand side limit if ∀ x n, and Definition: Left-hand side limit if ∀ x n, and Finite limit at finite point iff ∀ x n  x 0 and Definition: Let the function f is defined in some deleted environment of point x o. It is said that the limiting value of function f is A at the point x 0, if for all sequences then the sequence of functions value. The sequence f(x n ) tends to the same limiting value for all independent variable sequences x n which tends to the value x o independently from that the function f is defined or not at the value x=x o. With using symbols: Remark: finite limit right-hand and left –hand limits at x o and these are equal. Relationships between the operations and the limiting values : Theorem. If then ha B ≠ 0 PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

A notable limit y=1∕x y=-1∕x PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

II. Infinite limit at finite point Consider function Examine function f at the environment of point x o =2. Consider x n →2 but x n ≠ 2  Definition: Let the function f is defined at some deleted environment of point x o. It is said that the limiting value of function f is +∞ at the point x 0, if arbitrary sequence but then Remark: Consider Therefore the limit doesn’t exist at the point x o =2. PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

III. Finite limit at infinity Examine the properties of functions, when x tends to infinity. Consider,then Defintion: Let the function f is defined over the unbounded interval x > a. The limiting value of function f(x) is A as x tends to +∞, if then With using symbols: Remark: PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

IV. Infinite limit at infinity Examine the functions, if the value x is increasing beyond all bounds. Consider, then Definition: Let the function f is defined over the unbounded interval x > a.. The limiting value of function f(x) is +∞ at +∞, if as With using symbols: Remark: not defined PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

Continuity at a point Definition: The function f(x) is continuous at point x o, if 1) the function is defined at point x o and near at some environment of x o 2) 3) Visually: g(x) is continuous at point x o =1 because of we don’t lift up the pencil when drawing the graph of function at x o =1. What requirements have to fulfill g(x) at the point x o =1? PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

Definition: A function f(x) is continuous right-handed at point x o, if 1) function is defined at point x o and on their right-handed environment 2) right- handed limit is finite 3) Function is not continuous at x 0 =0 because of only the right-hand side limit exits. One-sided continuity, types of discontinuity If a function is not continuous at a point of the domain of definition then it is called discontinuous at that point. Removable discontinuity : if the finite limit exists at point x 0 Not removable discontinuity: if dosn’t exist the finite limit at the point x 0 PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

Theorem: If functions f(x) and g(x) is continuous at point x o, then their sum, substract and product is also continuous at point x o. Their quotient is also continuous when the denominator is not zero at the point x 0. Theorems on continuity Theorem: If the function f(x) is continuous at the point x o and exists their inverse function then the inverse is continuous at the point f(x o ). Continuity of elementary functions Theorem: The elementary functions f(x)=x, f(x)=e x, f(x)=sin x are continuous at all points. (The continuity of all others elementary functions follows from the previous theorems.) Proof: Theorem: If the function g(x) is continuous at the point x o and the function f(x) is continuous at point g(x o ) then the composition function f[g(x)] is also continuous at point x o. therefore (limiting value = substitution value) PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó

Definition: The function f(x) is said to continuous on the closed interval [a,b ] if it is continuous in all interior points of the interval and right-continuous at point a, left-continuous at point b. Properties of continuous functions on closed interval II. Theorem: (Weierstrass). Suppose that a function is continuous on the closed interval [a,b]. Then there exist the maximum value and the minimum value of the function in the closed interval [a,b]. III. Theorem: (Bolzano) If a function f is continuous on the closed interval [ a,b ] then it takes any value between f(a) and f(b). I. Theorem: A continuous function on closed interval [a,b] is bounded on [a,b]. Remark: It is essential assumption that the interval is closed. For example function f(x)=1/x is continuous on the open interval (0,1) and discontinuous at the left end point x 0 =0 so it is discontinuous and unbounded on [0,1]. PTE PMMIK Mathematics Department Perjésiné dr. Hámori Ildikó