The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. For example, given the function f (x) = 3x , you could say, “The limit of f (x) as x approaches 2 is 6 .” Symbolically, this is written f (x) = 6 . In the following sections, we will more carefully define a limit, as well as give examples of limits of functions to help clarify the concept.
Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. Using limits, we’ll learn a better and far more precise way of defining continuity as well. With an understanding of the concepts of limits and continuity, you are ready for calculus.
– A function is continuous at a point if the three following conditions are met: 1) f (a)
is defined. 2)f (x)
exists. 3)f (x) = f (a)
. A conceptual way to describe continuity is this: A function is continuous if its graph can be traced with a pen without lifting the pen from the page.
Infinite Discontinuity – A category of discontinuity in which a vertical asymptote exists at x = a and f (a) is undefined.
– A category of discontinuity in whichf (x)≠f (x)
, but both of these limits exist and are finite.
Limit – The value A to which a function f (x) gets arbitrarily close as the value of the independent variable x gets arbitrarily close to a given value a . Such a limit is symbolized this way: lim xâÜ’a f (x) = A .
– A limit based entirely on the values of a function taken at an x
-value slightly greater than or less than a given value. Whereas a two-sided limitf (x)
takes into account the values of x
which are both greater than and less than a
, a one-sided limit from the leftf (x)
or from the rightf (x)
takes into account only values of x
less than a
, or greater than a
Point Discontinuity – A category of discontinuity in which a function has a well-defined two-sided limit at x = a , but either f (x) is not defined at a or its value at a is not equal to this limit.