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The arctangent function, often denoted as \(\tan^{-1}(x)\), is an inverse trigonometric function. It serves the purpose of indicating the angle whose tangent is \(x\). A prominent feature of the arctangent function is its horizontal asymptotes. As \(x\) approaches infinity, \(\tan^{-1}(x)\) approaches \(\frac{\pi}{2}\) radians, revealing its behavior over larger values.
This property makes it particularly appealing in limit problems. The function transforms seemingly complex trigonometric expressions into more manageable calculations. Understanding its asymptotic behavior is crucial in both practical and theoretical contexts.
Furthermore, the domain of \(\tan^{-1}(x)\) extends across all real numbers while its range is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). With this understanding, one can evaluate limits that involve the arctangent function with greater ease.

Trigonometric limits often involve functions like sine, cosine, and tangent and their inverses. These types of limits express the behavior of trigonometric functions as the input grows infinitely large or infinitesimally small.
In this exercise, the cosine function plays a key role. It cycles repeatedly between -1 and 1, while its period is \(2\pi\). By computing limits of cosine, solutions can reveal periodic behaviors or fixed outcomes, as seen in approaching values like \(\pi\).
Sometimes, combining trigonometric functions with limits might seem intimidating. However, by breaking down complex expressions into simpler parts (like computing \(\lim_{x \to +\infty} 2\tan^{-1}(x)\) before finding \(\cos(\pi)\)), students can tackle these problems step-by-step. It is also crucial to remember key limit properties, including continuity and periodicity.

Infinite limits describe the behavior of a function as the input grows boundlessly large. In the realm of trigonometric functions, infinite limits highlight how these functions level off or correspond to certain asymptotic values.
For instance, as \(x\) approaches infinity, the calculation \(\lim_{x \to +\infty} \tan^{-1}(x) = \frac{\pi}{2}\) signifies the leveling behavior of the arctangent function. Such insights can simplify the understanding of involved expressions by presenting predictable and manageable results.
When interpreting infinite limits, it’s important to grasp how different functions "behave" in vast or extremely small domains. These concepts extend far and wide in calculus, empowering mathematicians to handle large-scale growth or decay more effectively with tools such as horizontal asymptotes.