Video: Finding the Limit of a Rational Function at a Point by Evaluating the Function with Given Values

Determine lim_(π‘₯ β†’ βˆ’10) (βˆ’π‘₯Β² + 10 π‘₯)/(π‘₯Β² βˆ’ 100), if it exists, by evaluating the function at the following values of π‘₯: βˆ’9.5, βˆ’9.9, βˆ’9.95, βˆ’9.99, βˆ’9.999, βˆ’9.9999, βˆ’10.5, βˆ’10.1, βˆ’10.05, βˆ’10.01, βˆ’10.001, and βˆ’10.0001, rounding the result to the nearest six decimal places.

05:29

Video Transcript

Determine the limit of the function negative π‘₯ squared plus 10π‘₯ all divided by π‘₯ squared minus 100 as π‘₯ approaches negative 10, if it exists, by evaluating the function at the following values of π‘₯. Negative 9.5, negative 9.9, negative 9.95, negative 9.99, negative 9.999, negative 9.9999, negative 10.5, negative 10.1, negative 10.05, negative 10.01, negative 10.001, and negative 10.0001. Rounding the result. to The nearest six decimal places.

First, we noticed that the question is asking us to determine the limit of this function as π‘₯ is approaching negative 10. Then the question asks us to do this by evaluating this function of these values of π‘₯. Then if it is applicable, we will need to round our answer to the nearest six decimal places. We know that the limit of a function 𝑓 as π‘₯ approaches π‘Ž is equal to 𝐿 when both the left hand and right hand limit exist and are both equal to 𝐿. So in our question, we’re given this value as our function 𝑓. And we’re given that π‘₯ is approaching negative 10.

Now, since the question is asking us to determine this limit by evaluation, we can use a function table. So we make a table with our input values of π‘₯ on the top and our output values of 𝑓 of π‘₯ on the bottom. Since we’re going to use this function table to determine the limit as π‘₯ approaches negative 10, we will include an π‘₯-value at negative 10 in our table. But we will leave the output value blank. Next, we add in all the π‘₯-values given to us in the question. We take care to write these into our table in ascending order. So we start with negative 10, the most negative value. And we will end with negative 9.5.

Now, we want to fill in all the values for our outputs of 𝑓 of π‘₯. We do this by substituting in each value of π‘₯ in our table into the function 𝑓. And then we will evaluate the output. So when π‘₯ is negative 10.5, we’ll have the function’s output of negative 10.5 is equal to negative 10.5 all squared, multiplied by minus one plus 10 multiplied by negative 10.5 all divided by negative 10.5 squared minus 100. We can evaluate this to give us negative 215.25 divided by 10.25. We can then evaluate this as negative 21. This is the output of the function 𝑓 when π‘₯ is equal to negative 10.5. Therefore, we can fill in this value of negative 21 into our function table, here.

We can now do the same with the next value in our table, negative 10.1. We get this expression for the output of our function 𝑓 when π‘₯ is equal to negative 10.1. We can then evaluate this expression. And we find that it gives us a value of negative 101. And now, just like before, we can fill in this value of negative 101 into our function table, here. We can then follow the same process for every single π‘₯-value given in the question which will give us the following table. Now, since our definition requires that the limit from both the left-hand side and the right-hand side exist and are equal, we want to evaluate the left-hand limit and the right-hand limit, using our table. So we can evaluate the left-hand side limit by looking at these values in our table.

If we were to write these output values in a sequence, we would get the sequence negative 21, negative 101, negative 201, negative 1001, negative 10001, and then negative 100001. We can see that the sequence is getting progressively more negative. And therefore it does not converge. In fact, if we look at the outputs close to negative 10, we see that just a tiny increase in our input because the output decreased from around negative 10000 to around negative 100000. Again, this indicates that our limit from the left does not converge. We can use the same methods to analyze the right-hand limit by looking at these values in our table. We can write the outputs in the following sequence where we noticed that the respective inputs of the sequence are approaching negative 10.

We can see that each value in the sequence is getting larger and larger and does not seem to be converging to any value. This would indicate to us that the right-hand limit as π‘₯ approaches negative 10 does not exist. In fact, if we look at the outputs close to negative 10, we see that just a tiny decrease in our input equals the output increase from around 10000 to around 100000. Again, this indicates that our limit from the right does not converge. Since the output values from the left and right are not approaching the same value, we can conclude that the limit in our question does not exist.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.