Determine the following limits.
lim x→∞ (3 tan^-1 x + 2)

Verified step by step guidance

1. Recognize that the problem involves finding the limit of a function as \( x \) approaches infinity: \( \lim_{x \to \infty} (3 \tan^{-1} x + 2) \).
2. Recall the behavior of the inverse tangent function, \( \tan^{-1} x \), as \( x \to \infty \). The function approaches \( \frac{\pi}{2} \).
3. Substitute the asymptotic value of \( \tan^{-1} x \) into the expression: \( 3 \tan^{-1} x + 2 \approx 3 \left( \frac{\pi}{2} \right) + 2 \).
4. Simplify the expression by multiplying and adding the constants: \( 3 \times \frac{\pi}{2} + 2 \).
5. Conclude that the limit is the simplified expression from the previous step.

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine their end behavior. In this context, we analyze how the function approaches a specific value as x becomes very large.

Inverse Tangent Function

The inverse tangent function, denoted as tan^-1(x) or arctan(x), is a function that returns the angle whose tangent is x. As x approaches infinity, the value of arctan(x) approaches π/2. This property is essential for solving the limit in the question, as it helps us understand the limiting behavior of the function involved.

Constant Addition in Limits

When evaluating limits, adding a constant to a function does not affect the limit itself. This principle allows us to simplify the limit calculation by focusing on the behavior of the variable part of the function. In this case, after determining the limit of the arctan function, we can simply add 2 to find the final limit value.