WEBVTT
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Limits from Tables and Graphs
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In this video, we will learn how to evaluate the limit of a function using tables and graphs.
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Weβll be looking at a variety of examples of how we can use tables and graphs in order to evaluate limits of different functions.
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Using tables and graphs can be a very nice visual way to find a limit.
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But before we move on to the use of tables and graphs, letβs recall what a limit is.
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This here is a limit.
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We can say that it is the limit as π₯ approaches π of π of π₯.
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And when weβre considering this limit, what weβre actually thinking is the value π of π₯ approaches as π₯ gets closer and closer to π.
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Letβs start by discussing how we can use a table to find a limit such as this one.
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We will do this by considering the following example.
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Estimate the limit as π₯ tends to negative two of π of π₯ from the given table.
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As we can see from the table, weβve been given π₯-values getting closer and closer to negative two from above and from below.
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And weβve been given their corresponding π of π₯ values.
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Letβs start by considering the π₯-values from below negative two.
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We have that π of negative 2.1 is equal to 36.9.
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π of negative 2.01 is equal to 36.09.
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π of negative 2.001 is equal to 36.009.
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Here, our π₯-values are getting closer and closer to two.
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We need to consider whatβs happening to our π of π₯ values.
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We can quite clearly see that π of π₯ is getting closer and closer to 36.
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Letβs now consider the π₯-values just above negative two.
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We have that π of negative 1.9 is equal to 35.1.
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π of negative 1.99 is equal to 35.91.
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π of negative 1.999 is equal to 35.991.
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So again, here we can see that our π₯-values are getting closer and closer to negative two.
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As this is happening to π₯, we need to consider whatβs happening to π of π₯.
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Our π of π₯ values go 35.1, 35.91, and 35.991.
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Therefore, itβs safe to say that these π of π₯ values are tending towards 36.
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As we can see, as π₯ approaches negative two from both directions, the value which π of π₯ approaches agrees with one another.
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Theyβre both 36.
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Therefore, we can say that as π₯ tends towards negative two, π of π₯ approaches 36.
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And if we convert this into mathematical notation, we arrive at our estimate.
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And that is that the limit as π₯ approaches negative two of π of π₯ is equal to 36.
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Letβs now consider another example where we have to find the limit from a table, but with a slight difference.
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Determine the limit as π₯ tends to five of π₯ squared plus three π₯ over the square root of π₯ minus one by evaluating the function at π₯ is equal to 4.9, 4.95, 4.99, 4.995, 4.999, 5.001, 5.005, 5.01, 5.05, and 5.1, rounding to the nearest three decimal places.
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Now, our first step in answering this question is to evaluate the function at the given π₯-values.
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Now, the function we need to consider is the function inside the limit.
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And we can call this function π of π₯.
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Therefore, π of π₯ is equal to π₯ squared plus three π₯ over the square root of π₯ minus one.
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Letβs now draw a table of values for the π₯-values given in the question along with their corresponding π of π₯ values.
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And we can find these corresponding π of π₯ values by simply plugging their π₯ values into π of π₯.
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Substituting π₯ is equal to 4.9 into π₯ squared plus three π₯ over the square root of π₯ minus one, then we obtain that π of π₯ is equal to 19.602.
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Substituting in 4.95, we obtain that π of π₯ is equal to 19.800.
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We mustnβt forget to round our values of π of π₯ to three decimal places, since this is what the question has told us to do.
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Continuing this on, we find that π of 4.99 is 19.960.
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π of 4.995 is 19.980.
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π of 4.999 is 19.996.
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Continuing this on for the five remaining values, we get that π of 5.001 is 20.004.
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π of 5.005 is 20.020.
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π of 5.01 is 20.040.
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π of 5.05 is 20.200.
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And π of 5.1 is 20.402.
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Now, weβre evaluating the limit of π of π₯ as π₯ approaches five.
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The value of five comes between the values of 4.999 and 5.001.
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Letβs now look at the trend in the π of π₯ values as π₯ gets closer and closer to five.
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As π₯ approaches five from below, we can see that the π of π₯ values are getting closer and closer to 20.
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And if we look at the π of π₯ values as π₯ approaches five from above, then we can see that these π of π₯ values are also approaching 20.
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We know that they are approaching 20 since with each step that weβre getting closer to five with our π₯-values, weβre getting closer and closer to 20 with our π of π₯ values.
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However, we never actually reach 20 from either side.
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From this, we can say that as π₯ tends to five, π of π₯ approaches 20.
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If we convert this into math language, then we reach our solution.
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Which is that the limit as π₯ tends to five of π₯ squared plus three π₯ over the square root of π₯ minus one is equal to 20.
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Letβs now consider how we can find the limit of a function using a graph.
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Consider the following examples.
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If the graph represents the function π of π₯ is equal to π₯ minus three, determine the limit as π₯ tends to negative one of π of π₯.
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In order to find the limit of π of π₯ as π₯ approaches negative one, we simply need to look at the graph.
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However, we first need to know what weβre looking for.
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We can say that the limit as π₯ approaches negative one of π of π₯ is the value that π of π₯ approaches as π₯ tends to negative one.
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We need to consider the graph of π of π₯ around π₯ is equal to negative one.
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We can see that just to the right of the function, as π₯ approaches negative one, π of π₯ approaches negative four.
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And now, we can consider π of π₯ just to the left of negative one.
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We can see that the function is doing the same, but in the opposite direction.
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It is tending towards negative four.
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Therefore, using a wordy definition of a limit, we can say that the limit as π₯ approaches negative one of π of π₯ is equal to negative four.
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Now, weβve seen a relatively straight forward example of how we can use a graph to find a limit.
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Letβs consider a slightly less obvious example.
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Determine the limit as π₯ tends to two of the function represented by the graph.
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Now, if we look at the graph, we can see that the function is called π of π₯.
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And weβve being asked to find the limit of π of π₯ as π₯ tends to two.
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In other words, this is the limit as π₯ tends to two of π of π₯.
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This can also be described as the value π of π₯ approaches as π₯ tends to two.
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So we need to find this value of π of π₯ using our graph.
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We can consider what π of π₯ is doing around the value of π₯ is equal to two.
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Weβll need to consider π of π₯ on both the left and right of two.
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Letβs consider π of π₯ on the right of π₯ is equal to two.
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We can see that as π₯ gets closer and closer to two, π of π₯ is decreasing.
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And it is decreasing towards this point here, which has a value of three.
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So we can say that as π₯ tends to two from the right, the value of π of π₯ approaches three.
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Letβs now consider what happens on the left of π₯ is equal to two.
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We can again see that as π₯ gets closer and closer to two, the value of π of π₯ is decreasing.
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And from the graph, we can see that it is decreasing towards the same value that π of π₯ is approaching from the right as π₯ approaches two.
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And thatβs a value of three.
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So now we can say that as π₯ approaches two from the left, π of π₯ approaches three.
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Since π of π₯ approaches the same value from the left and the right of two, we can therefore conclude that the value that π of π₯ approaches as π₯ tends to two is three.
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And so, we reach our solution, which is that the limit as π₯ approaches two of π of π₯ is equal to three.
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In this last example, weβve seen how we can use a graph in order to find the value of a limit at a point, even if the graph has a sharp turn at that point.
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Letβs now move on to another example.
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Determine the limit as π₯ approaches two of π of π₯ if it exists.
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Here, we have been given the graph of π of π₯, and weβre trying to find the limit of π of π₯ as π₯ tends to two.
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If we try to find the value of π of π₯ when π₯ is equal to two, we can see that π is, in fact, undefined.
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However, this does not mean we cannot find the limit.
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We know that the limit as π₯ approaches two of π of π₯ is the value π of π₯ approaches as π₯ tends to two.
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In order to find this limit, we simply need to consider π of π₯ around two not specifically at two.
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Letβs consider the π₯-values just to the right and just to the left of π₯ is equal to two.
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Letβs start by looking at π of π₯ just to the left of π₯ is equal to two.
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We can see that as π₯ gets closer and closer to two from below, the value of π of π₯ is getting closer and closer to three.
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And if we consider the π₯-values just to the right of π₯ is equal to two, then we can see that as π₯ gets closer and closer to two from the right, the value of π of π₯ is decreasing and getting closer and closer to three.
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Since π of π₯ is tending to the same value as π₯ approaches two from both the left and right and this value is three, we can conclude that the limit as π₯ approaches two of π of π₯ is equal to three.
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In this last example, weβve seen how we still may be able to find the limit of π of π₯ as π₯ approaches a particular π₯-value, even if π is undefined at that particular π₯-value.
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We will consider one final example.
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Using the graph shown, determine the limit as π₯ tends to three of π of π₯.
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Here, we have the graph of π of π₯.
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And weβve been asked to find the limit as π₯ tends to three.
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We can see that at π₯ is equal to three, π of π₯ is defined to be negative five.
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However, when we are finding the limit of a function at a particular point, the value of that function at that point does not matter.
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What matters is whatβs happening to the function around that point.
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This is because the limit as π₯ approaches three of π of π₯ is defined to be the value π of π₯ approaches as π₯ tends to three.
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Letβs consider whatβs happening to π of π₯ to the left and to the right of π₯ is equal to three.
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If we consider π of π₯ to the left of π₯ is equal to three, we can see that π of π₯ is increasing and getting closer and closer to the value of two, as π₯ is getting closer and closer to three.
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And as π₯ approaches three from the right, π of π₯ is again increasing.
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And it is also getting closer and closer to the value of two.
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Now, this tells us what we need to know about π of π₯ as π₯ tends to three, since from both the left and right, the value of π of π₯ approaches two as π₯ approaches three.
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And so, even though the value of π of three is equal to negative five, the limit as π₯ approaches three of π of π₯ is equal to negative two, which is our solution to this question.
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In this previous example, weβve seen how even though a function may be defined at a different point at a particular π₯-value, the limit as π₯ approaches that particular π₯-value of π of π₯ may be different to the value of π of π₯ at that point.
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We have now covered a variety of examples.
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Letβs look at some key points of the video.
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Key Points
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The limit as π₯ approaches π of π of π₯ is the value π of π₯ approaches as π₯ tends to π.
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When finding a limit using a table, we consider the π of π₯ values as π₯ gets closer and closer to π from both above and below.
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The value π of π₯ approaches is equal to the limit as π₯ approaches π of π of π₯.
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When finding the limit as π₯ approaches π of π of π₯ from a graph, we consider the values of π of π₯ near π to find the value π of π₯ approaches as π₯ tends to π.
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This value is equal to the limit as π₯ tends to π of π of π₯.
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One quick final point here is, using tables and graphs, especially graphs, is a very visual way to find limits.
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And it can really help you to understand what a limit of a function is.
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If a question has asked you to find the limit of a function and not given you a graph of the function, you may find it useful to draw a graph.
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And then, you should easily be able to spot the limit of the function at that point.