Evaluate the following limit: lim x → ∞ tan ( x ) − 6 \lim_{x\rarr\infty}\tan\left(x\right)-6 .

give this problem a try. So here we're asked to evaluate the following limit, where we have the limit as x approaches infinity for the tangent of x minus 6. Now to solve this problem, what we need to do is first focus on where the x's are in this problem. I can see that the only x we have shows up inside the tangent function. So because of this, this is the portion we need to focus on. Now for figuring out the limit with when x approaches infinity for the tangent of x, I could draw out a table of values if I wanted to, but I think it might be better if we think about what a graph would look like for the tangent. So what I'm going to do is just kinda draw a rough graph of what I remember the tangent looking like here. And this is something that you may wanna recall your notes from trigonometry on. Now if we take a look at this graph right here, let's say that we have this b zero. And since we have a trigonometric function, I'm going to go by pi. So we're gonna have pi here, and then we'll have 2 pi. And then we'll go back on our graph to negative pi and then say negative 2 pi. And what the function is going to look like is something like this. You're going to have your tangent function, which is kind of these curves, and this is what the graph looks like. Now what I'm taking a look at here is how the function is going to behave as x gets infinitely big. And if we keep going to the right on our x axis, this pattern is not going to change for the tangent function. If we keep going to the right on the x axis here by the way, this is the x axis, that's the y. If we keep going to the right, these curves are just gonna keep going. We're never going to see this pattern kind of break between this with this function. So we don't really know where we're gonna end up. We could end up somewhere here. We could end up somewhere there. We could end up somewhere there. We don't really converge on a specific value. It kinda just keeps going up and down and and kinda between these these curves that we have. And because of this, we would see that the tangent of x here does not exist. So when we have the limit as x for superinte, this whole thing does not exist. And then we still have this minus 6, but we subtract 6 from something that does not exist, the entire answer is just not going to exist. So this right here would be the solution to this problem. We we don't really have an answer that we converge on. So that is the main idea and how you can solve these types of problems when you're dealing with limits as x approaches infinity. Hope you found this video helpful, and let's move on.

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23. Intro to Derivatives & Area Under the Curve

Limits at Infinity

Precalculus

23. Intro to Derivatives & Area Under the Curve

Limits at Infinity

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Multiple Choice

Evaluate the following limit: $\lim_{x \to \infty} \tan (x) - 6$ .

-6

Does not exist

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Verified step by step guidance

Understand the behavior of the tangent function: The tangent function, $ \tan(x) $, is periodic with a period of $ \pi $. It has vertical asymptotes at $ x = \frac{\pi}{2} + n\pi $ for any integer $ n $, where it approaches $ \infty $ or $ -\infty $.

Consider the limit $ \lim_{x \to \infty} \tan(x) - 6 $: As $ x \to \infty $, $ \tan(x) $ does not approach a single value because it oscillates between $ -\infty $ and $ \infty $ due to its periodic nature.

Subtracting 6 from $ \tan(x) $: The expression $ \tan(x) - 6 $ will also oscillate between $ -\infty $ and $ \infty $ as $ x \to \infty $, since subtracting a constant does not affect the unbounded oscillation.

Conclude about the limit: Since $ \tan(x) - 6 $ does not approach a single finite value as $ x \to \infty $, the limit $ \lim_{x \to \infty} (\tan(x) - 6) $ does not exist.

Summarize the result: The limit does not exist because the function $ \tan(x) - 6 $ continues to oscillate indefinitely as $ x \to \infty $.