What happens if the limit is infinity

It gives no indication that the respective limits are 1 and 2. That is why this expression is called indeterminate. Rather, keep in mind that we are taking limits. What is really happening is that the numerator is shrinking to 0 while the denominator is also shrinking to 0.

The respective rates at which they do this are very important and determine the actual value of the limit. An indeterminate form indicates that one needs to do more work in order to compute the limit. That work may be algebraic such as factoring and canceling or it may require a tool such as the Squeeze Theorem. In a later section we will learn a technique called l'Hospital's Rule that provides another way to handle indeterminate forms.

Again, keep in mind that these are the "blind'' results of evaluating a limit, and each, in and of itself, has no meaning. Graphically, it concerns the behavior of the function to the "far right'' of the graph.

We make this notion more explicit in the following definition. Definition 6: Limits at Infinity and Horizontal Asymptote.

Horizontal asymptotes can take on a variety of forms. Figure 1. Doing this, we get. This procedure works for any rational function. In fact, it gives us the following theorem. We can work out the sign positive or negative by looking at the signs of the terms with the largest exponent , just like how we found the coefficients above:. This formula gets closer to the value of e Euler's number as n increases :. So instead of trying to work it out for infinity because we can't get a sensible answer , let's try larger and larger values of n:.

Yes, it is heading towards the value 2. I have taken a gentle approach to limits so far, and shown tables and graphs to illustrate the points. But to "evaluate" in other words calculate the value of a limit can take a bit more effort.

Find out more at Evaluating Limits. Answer: We don't know! So when we say that the limit is infinity, we mean that there is no number that we can name. The student should be aware that the word infinite as it is used and has been used historically in calculus, does not have the same meaning as in the theory of infinite sets. See this from Wikipedia , especially the views of Carl Friedrich Gauss in the section "Reception of the argument. We say that a variable "becomes infinite" or "tends to infinity" if, beginning with a certain term in a sequence of its values, the absolute value of that term and of any subsequent term we name is greater than any positive number we name, however large.

When the variable is x and takes on only positive values, then x becomes positively infinite. We write. If x takes on only negative values, it becomes negatively infinite, in which case we write. In both cases, we mean: No matter what large number M we name, we get to a point in a sequence of values of x that their absolute values become greater than M.

When the variable is a function f x , and it becomes positively or negatively infinite when x approaches the value c , then we write. Although we write the symbol "lim" for limit, those algebraic statements mean: The limit of f x as x approaches c does not exist.

Again, a limit is a number. Definition 2. Definition 4 is the definition of "becomes infinite;" it is not the definition of a limit. That symbol by itself has no meaning. Let us see what happens to the values of y as x approaches 0 from the right:. As the sequence of values of x become very small numbers, then the sequence of values of y , the reciprocals, become very large numbers.

The values of y will become and remain greater, for example, than 10 If x approaches 0 from the left , then the values of become large negative numbers. In that case, we write. Topic 18 of Precalculus. Next, let us consider the case when x becomes infinite , that is, when its values become large positive numbers to the extreme right of 0. In that case, becomes a very small number, namely 0. We should read that as "the limit as x becomes infinite," not as " x approaches infinity" because again, infinity is neither a number nor a place.

On the other hand, we could read that however we please "the limit as x becomes dizzy" , as long as whatever expression we use refers to the condition of Definition 4.