{"id":332,"date":"2017-08-28T04:31:12","date_gmt":"2017-08-28T04:31:12","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=332"},"modified":"2018-03-01T08:50:24","modified_gmt":"2018-03-01T08:50:24","slug":"limits-and-continuity","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/limits-and-continuity\/","title":{"rendered":"Limits and Continuity"},"content":{"rendered":"
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Workshop Index<\/a>\u00a0Next\u00a0Lesson \u27a1<\/a><\/div>\n

Limits and Continuity<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study limits of functions and the continuity of functions as a prerequisite to understanding differential calculus.<\/p>\n

What is the limit of a function?<\/h3>\n

The limit of a function<\/abbr> is the value of the function as its variable approaches a number. The expression <\/em> is read the limit of as x approaches a is B<\/i>. Note that must be defined everywhere along an open interval around a<\/i> (which does not necessarily include a<\/i>).<\/p>\n

An example of a formal definition for a limit is:
\nFor every number in an open interval around a<\/i> (except perhaps a<\/i> itself), the limit exists if, for any , there exists such that . <\/span><\/p>\n

In other words, approaches the limit B<\/i> as x<\/i> approaches a<\/i>, if the absolute value of the difference between and B<\/i> can be made smaller and smaller as x<\/i> approaches a<\/i>.<\/p>\n

There is a subtle, but very important detail within this definition. When we refer to an open interval around a<\/i>, it is important to note that an open interval around a point a<\/i> contains, by definition, points on either side of a<\/i>. Because we only have one limit B<\/i> for each a<\/i> in the domain of our function, B<\/i> must have the same value whether we approach a<\/i> from the left or right side.<\/p>\n

The simplest method to compute the limit of a function, whenever possible, is to plug in the value of a<\/i> and see if the limit describes a real value. For example, what is ? Note that the function is not defined at since that would lead to division by zero. However, we are interested in the open interval around , so simply substituting this value of x<\/i> will work.<\/p>\n

<\/p>\n

Just because this function is not defined at , does not mean that this limit does not exist as . The limit still exists. This can be proven in several ways. One method is to substitute values close to and on either side of \u20133 and see what results. This method is not a formal proof, but it is convincing enough for simple functions like these. Using a calculator, compute the value of the limit at and . The results are \u20131.001 and \u20130.999, respectively. As you try values even closer to \u20133, your will see that your answers get even closer to \u20131.<\/p>\n

A second method is to graph the function.<\/p>\n

\"Graph<\/p>\n

A third method is to simplify the function, when possible. Note that can be factored as . Therefore, the limit can be rewritten<\/p>\n

<\/p>\n

Remember from algebra that you can cancel out similar terms in the numerator and denominator, as long as the value of the denominator is never zero. As you can see, the denominator is zero at . We are looking at values as approaches<\/i> \u2013 3, but never at . Therefore, it is safe to cancel these terms.<\/p>\n

<\/p>\n

Unlike the plug-in method described above, this method serves as proof that the limit is indeed \u2013 1.<\/p>\n

Now, let\u2019s consider . At , we see that . Does this mean that ?
\nAccording to our definition of a limit, the answer is no. The function does not have to be defined at for the limit to exist. It must exist in an open interval around . While that is true for (that is, to the right<\/i> of zero), we know that the function
\nis not defined for (to the left<\/i> of zero), since the result is the square root of a negative number.<\/p>\n

A one-sided limit<\/abbr> exists if is restricted in one direction but not the other. For this example, if <\/span>x<\/span><\/em> is restricted on the right, we say that we have a right-hand limit<\/i> or a one-sided limit from the right. This is written<\/span> . A left-hand limit<\/i> or a one-sided limit from the left<\/i> is written .<\/p>\n

\n

Question<\/h4>\n
\n

Consider the piecewise function . Which statement is true for this function?<\/p>\n

    \n
  1. <\/p>\n<\/li>\n
  2. is not defined<\/li>\n
  3. <\/li>\n
  4. <\/li>\n<\/ol>\n<\/div>\n

    Reveal Answer <\/a><\/p>\n

    \n

    The correct choice is B. Although at , the value of the function changes in the open interval to the left and right of 0. Therefore, B is true. Choices C and D are reversed; they would be correct if the signs of their limits were swapped, or if the signs above their zeroes were swapped.<\/p>\n<\/div>\n<\/section>\n

    Let\u2019s look at the functions and . Below are their graphs:<\/p>\n\n\n\n\n
    \n

    \"Graph<\/p>\n<\/td>\n

    		\n

    \"<\/p>\n<\/td>\n<\/tr>\n

    f(x)<\/td>\ng(x)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    We see from the graph of that, as approaches 3 from either the right or left, <\/span> blows up<\/span><\/em>, or approaches infinity. This is written <\/span><\/p>\n

    \n and .\n<\/p>\n

    In this case, we say that increases without bound<\/i>. Since this is true for the open interval around 3 (but not including ), we can write . This is an example of an infinite limit<\/abbr>. This does not mean the limit exists but gives us a more descriptive way of explaining why the limit doesn’t exist. Note that if we substitute , this limit approaches as approaches 3. In this case, we say that the limit decreases without bound.<\/i><\/p>\n

    One-sided limits can also be infinite limits, as seen by . From the graph, it is clear that and . The limit decreases without bound to the left of , and increases without bound to the right of .<\/p>\n

    Not to be confused with an infinite limit, there is also a limit at infinity<\/abbr>. The limit as approaches infinity (or negative infinity) exists if the value of approaches B<\/i> as increases (or decreases) without bound. Limits at infinity are written .<\/p>\n

    Consider the function . Below is a graph of this function.<\/p>\n

    \"Graph<\/p>\n

    (a) Is there an infinite limit to this function for some value of x<\/i>?<\/p>\n

    (b) Is the limit at x <\/i>= 4 one-sided or two-sided?<\/p>\n

    (c) What are the limits at infinity (positive and negative)?<\/p>\n

    (a) Yes, as x<\/i> approaches 4 from the left and right, the function blows up negatively and positively, respectively.<\/p>\n

    (b) Since the limits to the left and right of are different, the two-sided limit does not exist. We can write the one-sided infinite limits as and .<\/p>\n

    (c) . The asymptote at y<\/i> = 5 is an indication of this limit at infinity. To better understand this answer, note that, as becomes very large, the denominator looks more and more like . (For example, if Bill Gates has $40 billion and loses $4 playing poker, he still essentially has $40 billion.) Therefore, as :<\/p>\n

    <\/p>\n

    As , the same condition holds.<\/p>\n

    What are the limit theorems?<\/h3>\n

    The following theorems involve the limits of sums, products, quotients, and compositions of functions. While they are very useful, keep in mind that these theorems are only true when the individual limits and both exist. These theorems will come in handy as we prove some important rules in differential calculus.<\/p>\n

    \n

    Important Tidbit<\/h4>\n