Can you do limits on a calculator

In addition to this, understanding how a human would take limits and reproducing human-readable steps is critical, and thanks to our step-by-step functionality, Wolfram Alpha can also demonstrate the techniques that a person would use to compute limits. Uh oh! Wolfram Alpha doesn't run without JavaScript. Please enable JavaScript. If you don't know how, you can find instructions here. Once you've done that, refresh this page to start using Wolfram Alpha.

Function to find the limit of:. Similarly, we could consider what the behavior is of f x when x decreases without bound, and if f x were to approach a specific value L , we could write something like this:. Now we move on to some examples where we consider the behavior of f x as x increases or decreases without bound. In other words, we try to determine if f x approaches a specific fixed finite value as we go farther and farther to the right, or the left, on the graph of f x.

For the function in the graph below, we first consider the behavior of f x as as x increases without bound, or in other words, we consider what happens to f x as we move farther and farther to the right on the graph. In this case, f x appears to get closer and closer to zero.

It can never reach zero, because the function has no end: x can continue to increase forever. But the behavior of the function as x increases is that it grows ever closer to 0, even if it can never reach it. Similarly, as x decreases without bound, or as we move farther and farther to the left on the graph, f x appears to get closer and closer to zero. Again, here the behavior of f x as x decreases or grows more and more negative is that it grows ever closer to 0, even if it can never reach it.

In this case, f x appears to get closer and closer to two. It can never reach two, because the function has no end: x can continue to increase forever. But the behavior of the function as x increases is that it grows ever closer to 2, even if it can never reach it.

Similarly, as x decreases without bound, or as we move farther and farther to the left on the graph, f x appears to get closer and closer to two. Again, here the behavior of f x as x decreases or grows more and more negative is that it grows ever closer to 2, even if it can never reach it.

In this case, f x appears to increase without bound: it just seems to get bigger and bigger as we move to the right on the graph, without ever approaching a specific y -value. Similarly, as x decreases without bound, or as we move farther and farther to the left on the graph, f x appears to decrease without bound: it just seems to get smaller and smaller or more and more negative as we move to the left on the graph.

We note that it need not be the case that f x increases without bound as x increases without bound and that f x decreases without bound as x decreases without bound. For example, for the function in the graph below, we would have the opposite:. It is also possible for a function to have limits at positive infinity and at negative infinity that are different or even for the limit to exist for one of these, but not for the other.

All this statement really means is that the behavior of f x could be very different on the far left of our graph than on the far right. For example, in the graph below, we can see that a limit does exist both as x decreases without bound and as x increases without bound, and that this limit is different in each case:.

There is no reason why our limit would need to be negative as x becomes "more negative" i. For example, we could have the opposite case, as we do in the function given in the following graph:. One final possibility when we look for a limit as x approaches c is that f x never approaches anything as x gets closer and closer to c : we could have behavior, like in the graph given below, where f x just oscillates more and more wildly as x gets closer and closer to c from either the left or the right.

With the picture of the graph above, it's hard to be sure exactly what is happening as x gets closer and closer to 1: it looks like the oscillations are getting more and more dense, so that f x is continuously moving back and forth between -1 and 1 without ever settling down, but it is impossible to be sure.

Without an actual equation to look at that would explain exactly what the values of f x are as we get closer to 1, we actually can't prove that this is the behavior of this particular function, so for now, I'll just ask you to take my word for it. But to get a better idea of how this behavior of f x really works as x approaches 1, you can experiment a bit with the interactive animation below.

In the interactive animation below, you can see this behavior more clearly by moving the slider to the right, which zooms in the x -values on the graph around one. We can see that as we zoom in around one, the graph just oscillates more and more frequently, until it is so dense that we can no longer see the spaces between the graph and the blank space around it.

In this case, the only thing that we can write is simply that:. In this case, we can NOT write that the limit is equal to infinity, because the behavior of f x as x approaches 1 is NOT that it increases without bound - rather, the behavior of f x as x approaches 1 is to oscillate indefinitely among -1, 1, and all the numbers that fall in between.

This is important, because it is essential for us to understand that limits may fail to exist for different reasons. In the example above we said the limit was 2 because it looked like it was going to be. But that is not really good enough!

The first thing to try is just putting the value of the limit in, and see if it works in other words substitution. For some fractions multiplying top and bottom by a conjugate can help. By finding the overall Degree of the Function we can find out whether the function's limit is 0, Infinity, -Infinity, or easily calculated from the coefficients.