So, each point on satisfies the criteria of continuity at a point, and our function is indeed continuous on the entire interval. Additionally, it is pretty obvious from the graph that the limit as x approaches some value c on the interval is equal to f(c). Clearly the limit as x approaches each point in is defined and f(x) is defined for every point in the interval.
![calculus limits calculus limits](https://3.bp.blogspot.com/-TtOsWvaLYQY/XBdwIMwRETI/AAAAAAAACjI/mhvDehh3xbgg1L2puAswZLHrlkRDpTlQACLcBGAs/s1600/limit.jpg)
That’s a lot of points - in fact that an infinite number of points! But, if we think about the points in general, we can make this a little easier. We already determined that f(x) is continuous at x = 2, and we can do the same thing for all the other points on, but… Since these are equal, we have satisfied all the conditions for continuity at a point thus, our function is continuous at x = 2.į) For a function to be continuous on an interval, it must be continuous on every point in that interval. We found in part (c) that the limit as x approaches 2 of f(x) equals 0 and that f(2) equals 0 as well. Now we only need to make sure that they are equal to each other. We have already established that our two-sided limit is defined and that f(c) is defined. However, you will see that some people do include that as part of the criteria for determining continuity. Note: f(c) should be defined, but I generally do not include that in my definition, because I feel as if it is pretty intuitive. However, we should prove this assertion, for what if we weren’t given a graph but rather an equation?įirst let’s think back to what our criteria for determining continuity at a point was: One might look at the graph given and intuitively decide that it is continuous at x = 2. Make sure that you did not use limits to find this value - recall that the limit as x approaches some c of f(x) does not necessarily equal f(c).Į) The function is certainly continuous at x = 2.
![calculus limits calculus limits](https://i.pinimg.com/originals/14/42/0c/14420ca99860064cf611972ad97a6fff.jpg)
Remember that if two one-sided limits of a function at some x-value exist and are equal, we can say that our overall limit exists and is equal to the same value as both one-sided limits.ĭ) Looking at the graph, f(2) = 0. Looking at the graph above, we see that f(x) gets super close to a value of 0 when x approaches 2 from the left.Ĭ) Yes! Notice that our left- and right- sided limits are equal. Looking at our graph, our function f(x) gets super close to a value of 0 when x approaches 2 from the right.ī) Similarly, we are looking at the limit as x approaches 2 of f(x), but this time it’s from the left. Enjoy! #1 Use the graph below of f(x) to answer the questions.Ī) The “+” superscript next to the 2 tells us that we are looking at the limit as x approaches 2 from the right of f(x). These problems vaguely range from easy to hard.
![calculus limits calculus limits](https://showme1-9071.kxcdn.com/2020/10/01/21/8Zjmrc8_ShowMe_last_thumb.jpg)
Here, we will apply those skills to few practice questions.Īttempt the problems on your own at first however, if you get stuck, the solution to each problem is just below it. In the last two articles, we talked about limits and their application in determining the continuity of a function.