a particle moves along x-axis so that its position at any t>=0 is given by x=arctant. What is the limiting position of the particle as t approaches infinity?
Ah, the infinite antics of the particle! Let's solve this equation and find out where it ends up.
Given x = arctan(t), we want to find the limiting position as t approaches infinity.
As t grows larger and larger, the arctan(t) function approaches π/2. So, the limiting position of the particle as t approaches infinity is x = π/2.
It seems our particle is a master at performing a never-ending balancing act! Enjoy the show!
To find the limiting position of the particle as t approaches infinity, we need to evaluate the position function x = arctan(t) as t goes to infinity.
The arctan function takes values between -π/2 and π/2. As t approaches infinity, the arctan function will approach its limiting value of π/2.
Therefore, the limiting position of the particle as t approaches infinity is x = π/2.
Here's a step-by-step explanation of how to get this answer:
1. Start with the position function x = arctan(t).
2. Consider what happens to the arctan function as t approaches infinity.
3. The arctan function has a limiting value of π/2 as t approaches infinity.
4. Therefore, the particle's limiting position is x = π/2 as t approaches infinity.
you know that tan(x) -> ∞ as x -> π/2
so, what do you think?