Consider the graph of 𝑓 of 𝑥 equals two to the 𝑥 sin four 𝑥. Estimate the limit of 𝑓 of 𝑥 as 𝑥 approaches minus infinity. Estimate the limit of 𝑓 of 𝑥 as 𝑥 approaches infinity.
There are two parts of this question. Both parts require us to estimate the value of a limit from the graph. We start with the first part where we have to estimate the value of the limit of 𝑓 of 𝑥 as 𝑥 approaches minus infinity. So as 𝑥 gets smaller and smaller, more and more negative without bound, what value, if any, does 𝑓 of 𝑥 approach?
Let’s take a look at our graph, picking a point with a small but positive value of 𝑥. And we see that we have a small but positive value of 𝑓 of 𝑥 as well, the 𝑦-coordinate. As we let 𝑥 decrease approaching negative infinity, we can see that the value of 𝑓 of 𝑥 is oscillating, as we look to the left. But as 𝑥 continues to decrease, we can no longer see the small oscillations as they’re getting smaller and smaller. The graph appears to hug the 𝑥-axis closer and closer, asymptotically approaching the 𝑥-axis. And so, we estimate that the limit of 𝑓 of 𝑥 as 𝑥 approaches minus infinity is zero.
As 𝑥 approaches negative infinity — that is as 𝑥 gets smaller and smaller or more and more negative without bound, moving further and further left on our graph — the value of 𝑓 of 𝑥 which is represented by the 𝑦-coordinates of the points on our graph gets closer and closer to zero or approaches zero. And so, this is our estimate for the limit of 𝑓 of 𝑥 as 𝑥 approaches minus infinity.
Now, for the second part of the question — where we have to estimate the value of the limit of 𝑓 of 𝑥 as 𝑥 approaches infinity, so as 𝑥 gets greater and greater without bound — what value, if any, does 𝑓 of 𝑥 approach? Again, we pick a value of 𝑥 and find the corresponding point on our graph. And we see that its 𝑦-coordinate, representing 𝑓 of 𝑥 for this value of 𝑥, is small but negative.
We want to know about the limit as 𝑥 approaches infinity. So we allow 𝑥 to increase. And we see that 𝑓 of 𝑥 increases, but then decreases as 𝑥 increases further. Indeed, the graph is oscillating between large and small values of 𝑓 of 𝑥, large and small 𝑦-coordinates of the points on the graph. And this pattern continues, with the oscillations getting wilder and wilder, bigger and bigger in amplitude, as 𝑥 increases.
And so, we see that there is no single number that 𝑓 of 𝑥 is getting closer and closer to, or approaching, as 𝑥 approaches infinity. So the value of this limit can’t be any real number. And also, we can see that 𝑓 of 𝑥 isn’t just increasing without bound or decreasing without bound. And so, the value of this limit isn’t infinity or minus infinity.
The only conclusion we can draw is that the limit does not exist. 𝑓 of 𝑥 is not approaching any real number, nor is it approaching infinity or minus infinity. So we’re out of options.
We’ve answered both parts of this question by looking at the graph of the function. And so, we’ve had to assume that the trends we see on the graph continue for values of 𝑥 which are not on the graph. For example, we’ve assumed that as 𝑥 continues to decrease off the graph, the value of 𝑓 of 𝑥 doesn’t suddenly shoot up again. And we’ve also assumed that the amplitude of the oscillations doesn’t decrease as 𝑥 increases off the graph.
That’s why we have the verb estimate in both parts of our question. The answer we get from looking at the graph isn’t guaranteed to be right. But, it’s our best estimate. You might like to think about how the rule of the function, 𝑓 of 𝑥 equals two to the 𝑥 sin four 𝑥, justifies the values of the limits that we’ve estimated using the graph.