What is lim _{x tends to infinity} tan inverse 1/[x] . (Gif)

We are tasked with solving the limit of the expression as x tends to infinity:

lim (x → ∞) tan⁻¹(1/x)

Step 1: Analyze the behavior of the function inside the arctangent.
The function inside the arctangent is 1/x. As x approaches infinity, 1/x tends to 0 because:

lim (x → ∞) (1/x) = 0

Thus, we now need to find the limit of tan⁻¹(0), since the expression inside the arctangent tends to 0 as x increases.

Step 2: Evaluate tan⁻¹(0).
The inverse tangent function, tan⁻¹(x), returns the angle whose tangent is x. We know that:

tan⁻¹(0) = 0

Step 3: Conclusion.
Since 1/x tends to 0 as x approaches infinity, we conclude that:

lim (x → ∞) tan⁻¹(1/x) = 0

Thus, the limit is 0.