Find Limit. Lim_x Rightarrow Infinity Arctan (X^2 + 1/X + 1)
The Correct Answer and Explanation is :
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Problem:
Find the limit:
[
\lim_{x \to \infty} \arctan \left( x^2 + \frac{1}{x} + 1 \right)
]
Solution:
- Understand the behavior of the function:
The given expression involves (\arctan), which is a function with a horizontal asymptote at (\frac{\pi}{2}). Specifically:
[
\lim_{y \to \infty} \arctan(y) = \frac{\pi}{2}
]
So, as the argument inside the (\arctan) increases, the overall function approaches (\frac{\pi}{2}). - Simplify the expression inside (\arctan):
As (x \to \infty), the term (\frac{1}{x}) approaches zero, and the expression inside the (\arctan) becomes:
[
x^2 + \frac{1}{x} + 1 \approx x^2 + 1
]
This is because (x^2) dominates the (\frac{1}{x}) term as (x) grows larger. - Limit of the argument:
As (x \to \infty), the expression (x^2 + 1) becomes infinitely large. Hence, we can substitute:
[
\lim_{x \to \infty} \left( x^2 + \frac{1}{x} + 1 \right) = \infty
] - Final Limit:
Since the argument of the (\arctan) function approaches infinity as (x \to \infty), we can apply the limit property of (\arctan):
[
\lim_{x \to \infty} \arctan \left( x^2 + \frac{1}{x} + 1 \right) = \arctan(\infty) = \frac{\pi}{2}
]
Conclusion:
The limit of (\arctan(x^2 + \frac{1}{x} + 1)) as (x \to \infty) is (\frac{\pi}{2}).
This solution shows how the asymptotic behavior of the function simplifies the limit problem, and understanding the asymptote of the arctangent function is key to determining the final answer.