F Tends to Infinity as X Tends to Infinity F is Not Unidformly Continuous

The limit of a function The definition of the limit of a function A limit on the left (a left-hand limit) and a limit on the right (a right-hand limit) Continuous function Limits at infinity (or limits of functions as x approaches positive or negative infinity) Infinite limits The limit of a function examples Vertical, horizontal and slant (or oblique) asymptotes The definition of the limit of a function The limit of a function is a real number L that f (x) approaches as x approaches a given real number a , written if for any e > 0 there is a d ( e ) > 0 such that | f (x) - L | < e   whenever | x - a | < d ( e ) .

The definition says, no matter how small a positive number e we take, we can find a positive number d such that, for an arbitrary chosen value of x from the interval a - d < x < a + d , the corresponding function's values lie inside the interval L - e < f (x) < L + e , as shows the right figure. That is, the function's values can be made arbitrarily close to the number L   by choosing x sufficiently close to a , but not equal to a .

Therefore, the number d , that measures the distance between a point x from the point a on the x -axis, depends on the number e that measures the distance between the point f (x) from the point L on the y -axis.

Example: Given

whenever

A limit is used to examine the behavior of a function near a point but not at the point. The function need not even be defined at the point. A limit on the left (a left-hand limit) and a limit on the right (a right-hand limit) The limit of a function where the variable x approaches the point a from the left or, where x is restricted to values less than a , is written The limit of a function where the variable x approaches the point a from the right or, where x is restricted to values grater than a , is written If a function has both a left-handed limit and a right-handed limit and they are equal, then it has a limit at the point. Thus, if Continuous function A real function y = f (x)   is continuous at a point a if it is defined at x = a and that is, if for every e > 0 there is a d ( e ) > 0 such that | f (x) - f (a) | < e   whenever | x - a | < d ( e ) . Therefore, if a function changes gradually as independent variable changes, so that at every value a , of the independent variable, the difference between f (x) and f (a) approaches zero as x approaches a . A function is said to be continuous if it is continuous at all points. Limits at infinity (or limits of functions as x approaches positive or negative infinity) We say that the limit of f(x) as x approaches positive infinity is L and write , if for any e > 0 there exists N > 0 such that | f (x) - L | < e   for all x > N ( e ) . We say that the limit of f (x) as x approaches negative infinity is L and write , if for any e > 0 there exists N > 0 such that | f (x) - L | < e   for all x < - N ( e ) . Not all functions have real limits as x tends to plus or minus infinity. Thus for example, if f (x) tends to infinity as x tends to infinity we write if for every number N > 0 there is a number M > 0 such that f (x) > N   whenever x > M(N) . Infinite limits We write if f (x)  can be made arbitrarily large by choosing x sufficiently close but not equal to a . We write if f(x)  can be made arbitrarily large negative by choosing x sufficiently close but not equal to a . The limit of a function examples

As x tends to 0 from the right f(x) gets larger in positive sense. Therefore, Vertical, horizontal and slant (or oblique) asymptotes If a point (x, y)  moves along a curve f (x) and then at least one of its coordinates tends to infinity, while the distance between the point and a line tends to zero then, the line is called the asymptote of the curve. Vertical asymptote If there exists a number a such that then the line x = a   is the vertical asymptote. Horizontal asymptote If there exists a number c such that then the line y = c   is the horizontal asymptote. Calculus contents B Copyright � 2004 - 2020, Nabla Ltd.  All rights reserved.

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