# Video: Using Graphs to Estimate Limits at Infinity

Consider the graph of π(π₯) = 2^(π₯) sin (4π₯). Estimate lim_(π₯βββ) π(π₯). Estimate lim_(π₯ββ) π(π₯).

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### Video Transcript

Consider the graph of π of π₯ equals two to the π₯ sin four π₯. Estimate the limit of π of π₯ as π₯ approaches minus infinity. Estimate the limit of π of π₯ as π₯ 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 π of π₯ as π₯ approaches minus infinity. So as π₯ gets smaller and smaller, more and more negative without bound, what value, if any, does π of π₯ approach?

Letβs take a look at our graph, picking a point with a small but positive value of π₯. And we see that we have a small but positive value of π of π₯ as well, the π¦-coordinate. As we let π₯ decrease approaching negative infinity, we can see that the value of π of π₯ is oscillating, as we look to the left. But as π₯ continues to decrease, we can no longer see the small oscillations as theyβre getting smaller and smaller. The graph appears to hug the π₯-axis closer and closer, asymptotically approaching the π₯-axis. And so, we estimate that the limit of π of π₯ as π₯ approaches minus infinity is zero.

As π₯ approaches negative infinity β that is as π₯ gets smaller and smaller or more and more negative without bound, moving further and further left on our graph β the value of π of π₯ which is represented by the π¦-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 π of π₯ as π₯ approaches minus infinity.

Now, for the second part of the question β where we have to estimate the value of the limit of π of π₯ as π₯ approaches infinity, so as π₯ gets greater and greater without bound β what value, if any, does π of π₯ approach? Again, we pick a value of π₯ and find the corresponding point on our graph. And we see that its π¦-coordinate, representing π of π₯ for this value of π₯, is small but negative.

We want to know about the limit as π₯ approaches infinity. So we allow π₯ to increase. And we see that π of π₯ increases, but then decreases as π₯ increases further. Indeed, the graph is oscillating between large and small values of π of π₯, large and small π¦-coordinates of the points on the graph. And this pattern continues, with the oscillations getting wilder and wilder, bigger and bigger in amplitude, as π₯ increases.

And so, we see that there is no single number that π of π₯ is getting closer and closer to, or approaching, as π₯ approaches infinity. So the value of this limit canβt be any real number. And also, we can see that π of π₯ isnβt just increasing without bound or decreasing without bound. And so, the value of this limit isnβt infinity or minus infinity.

The only conclusion we can draw is that the limit does not exist. π of π₯ is not approaching any real number, nor is it approaching infinity or minus infinity. So weβre out of options.

Weβve answered both parts of this question by looking at the graph of the function. And so, weβve had to assume that the trends we see on the graph continue for values of π₯ which are not on the graph. For example, weβve assumed that as π₯ continues to decrease off the graph, the value of π of π₯ doesnβt suddenly shoot up again. And weβve also assumed that the amplitude of the oscillations doesnβt decrease as π₯ increases off the graph.

Thatβs why we have the verb estimate in both parts of our question. The answer we get from looking at the graph isnβt guaranteed to be right. But, itβs our best estimate. You might like to think about how the rule of the function, π of π₯ equals two to the π₯ sin four π₯, justifies the values of the limits that weβve estimated using the graph.