# what is infinity minus infinity in limits

Topics Login . It looks like we should have the following value for the right-hand limit in this case. We’ll also verify our analysis with a quick graph. If we did we would get division by zero. So, we have a positive constant divided by an increasingly small positive number. From this it’s easy to see that we have the following values for each of these limits. on the right). Now, let’s take a look at the left-hand limit. There are infinite number of infinite numbers, and they are not same. In this case we’re going to take smaller and smaller values of $$x$$, while staying negative this time. Recall from an Algebra class that a vertical asymptote is a vertical line (the dashed line at $$x = - 2$$ in the previous example) in which the graph will go towards infinity and/or minus infinity on one or both sides of the line. They will also hold if $$\mathop {\lim }\limits_{x \to c} f\left( x \right) = - \infty$$, with a change of sign on the infinities in the first three parts. The result will be an increasingly large and negative number. The result, as with the right-hand limit, will be an increasingly large positive number and so the left-hand limit will be. In all three cases notice that we can’t just plug in $$x = 0$$. But the first's limit is positive infinity, the second is negative infinity, and the third is zero. As with most of the examples in this section the normal limit does not exist since the two one-sided limits are not the same. So when we say that the limit is infinity, we mean that there is no number that we can name. Here’s a quick graph to verify our limits. First, notice that we can only evaluate the right-handed limit here. First, within the parenthesis, we subtract by reducing the common denominator and group terms in the numerator: We now remove the parenthesis by multiplying it by the term before it: When we can no longer operate, we replace the x with infinity and reach the infinite indeterminacy between infinity: To resolve this indeterminacy, we leave the term of highest degree and operate: Finally, we replace the x by infinite again, which is raised to less infinite by “e” than by properties of the powers, lower the denominator. So, in summary here are all the limits for this example as well as a quick graph verifying the limits. In this case then we’ll have a negative constant divided by an increasingly small negative number. Then. This website uses cookies to provide you with the best browsing experience. In this section we will take a look at limits whose value is infinity or minus infinity. This means that every time you visit this website you will need to enable or disable cookies again. Finally, since two one sided limits are not the same the normal limit won’t exist. So, we’re going to be taking a look at a couple of one-sided limits as well as the normal limit here. Let’s take a look at the right-handed limit first. for some real numbers $$c$$ and $$L$$. With this next example we’ll move away from just an $$x$$ in the denominator, but as we’ll see in the next couple of examples they work pretty much the same way. Also, as $$x$$ gets closer and closer to -2 then $$x + 2$$ will be getting closer and closer to zero, while staying positive as noted above. The right-hand limit should then be positive infinity. Tap to take a pic of the problem. A lot of people would say yes, but not really. Solved exercises of Limits to Infinity. Is infinity minus infinity zero? Likewise, we can make the function as large and negative as we want for all $$x$$’s sufficiently close to zero while staying negative (i.e. One way is to plug in some points and see what value the function is approaching. We know that the domain of any logarithm is only the positive numbers and so we can’t even talk about the left-handed limit because that would necessitate the use of negative numbers. So, we’ll have a numerator that is approaching a positive, non-zero constant divided by an increasingly small negative number. Not all infinite limits are the same. Section 2-6 : Infinite Limits. The limit is then, So, from our definition above it looks like we should have the following values for the two one sided limits. To see a more precise and mathematical definition of this kind of limit see the The Definition of the Limit section at the end of this chapter. Limit calculation with infinite indetermination minus infinity. Most problems are average. The answer will also be the division of the two largest variables -9/4, but don’t forget the minus sign. So, here is a table of values of $$x$$’s from both the left and the right. From this table we can see that as we make $$x$$ smaller and smaller the function $$\frac{1}{x}$$ gets larger and larger and will retain the same sign that $$x$$ originally had. Now that we have infinite limits under our belt we can easily define a vertical asymptote as follows. Rational Functions Reduce to a common denominator. Now, in this example, unlike the first one, the normal limit will exist and be infinity since the two one-sided limits both exist and have the same value. Now, there are several ways we could proceed here to get values for these limits. if we can make $$f(x)$$ arbitrarily large and negative for all $$x$$ sufficiently close to $$x=a$$, from both sides, without actually letting $$x = a$$. Is it infinity? Is it zero? Here are the official answers for this example as well as a quick graph of the function for verification purposes. Using these values we’ll be able to estimate the value of the two one-sided limits and once we have that done we can use the fact that the normal limit will exist only if the two one-sided limits exist and have the same value. Solutions are given WITHOUT the use of L'Hopital 's Rule cookie, we that! Cookies to provide you with the right-hand limit don ’ t deal with the right-hand limit we... To drop the absolute value bars in this case we ’ re going prove... Large positive number limit here absolute value bars in this case we have those we ’ ll attempt “! To infinity problems online with solution and steps of f ( x is. ” of infinity, but we do not know polynomial Functions Remove the common factor the... Taking a look at limits whose value is infinity looks like the limit., there are several ways we could proceed here to get from a quick sketch the... 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