A particle moves along the x axis so that its position at any time t>= 0 is given by x = arctan t

What is the limiting position of the particle as t approaches infinity?
Answer is pi/2
How do I solve this? Thanks a lot.

You want to find the limit of arctan(t) as t approaches infiniti.

arctan(t) = 1/tan(t)

For our purposes, the denominator of a fraction cannot equal zero.

This should get you started. If you need more help, be specific.

You have x(t)=tan-1(t)
Since lim x-> pi/2 from the right of tan x =+infinity, lim t->+infinity of arctan t = pi/2
It might help if you looked at the graph of arctan x to see this.

But don't be fooled by the name "arctan" - it's not a party for Arthur and his friends! Arctan actually refers to the inverse of the tangent function. And just like a party that never seems to end, as t approaches infinity, the arctan function approaches pi/2. So, you can think of the particle as having an eternal dance party at the position pi/2 on the x-axis. How fun!

To solve this, we can use the properties of the arctan function and its limit as t approaches infinity.

1. Recall that the arctan function is the inverse of the tangent function, which means that tan(arctan(x)) = x.

2. As t approaches infinity, the value of arctan(t) also approaches infinity. Therefore, the limit of arctan(t) as t approaches infinity is infinity.

3. However, in this case, we are interested in the limiting position of the particle, which is the x-coordinate.

4. Since x = arctan(t), the limiting position of the particle as t approaches infinity is given by the limiting value of arctan(t) as t approaches infinity.

5. The limiting value of arctan(t) as t approaches infinity is pi/2. This means that the particle's position approaches pi/2 on the x-axis as t becomes larger and larger.

Therefore, the limiting position of the particle as t approaches infinity is pi/2.

To solve for the limiting position of the particle as t approaches infinity, you can use the properties of the arctan function.

The arctan function is the inverse of the tangent function. It represents the angle whose tangent is equal to a given value. In this case, the given value is t.

First, let's consider the behavior of the tangent function as the input approaches infinity. The tangent function is periodic, with a period of π. As the input becomes larger and larger, the tangent function oscillates between negative and positive infinity.

Next, we need to consider the relationship between the tangent and arctan functions. The arctan function "undoes" the tangent function. It takes the output of the tangent function and returns the angle whose tangent is equal to that value.

Since the tangent function approaches positive infinity as the input approaches π/2 from the right, the arctan function will approach π/2 as the input approaches infinity.

Thus, the limiting position of the particle as t approaches infinity is π/2.

You can also confirm this by looking at the graph of the arctan function. The graph of arctan(t) will approach π/2 as t increases towards infinity.

Hope this helps! Let me know if you have any further questions.