Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value. If not possible, state the reason. As \(x \rightarrow-\infty,\) the value of arctan \(x \rightarrow\)

Step-by-Step Solution

Answer

The value of arctan \(x\) as \(x \rightarrow-\infty\) is \(-\pi/2\) or -90 degrees.

Step 1: Understand behavior of the arctan function

It is known that the arctan function is increasing, continuous and bounded between \(-\pi/2\) and \(\pi/2\), as the argument x ranges from negative infinity to positive infinity. This means that as the argument goes towards negative infinity, the arctan function should converge to one of its lower bounds.

Step 2: Identify the lower bound

From the range of the arctan function, it can be inferred that the lower limit for the arctan function as \(x \rightarrow -\infty\) is \(-\pi/2\) or -90 degrees.

Step 3: Confirm the result

Referencing the properties of the arctan function, it is confirmed that as x approaches negative infinity, arctan \(x \rightarrow -\pi/2\).

Key Concepts

Arctan Function

The arctan function is an inverse trigonometric function, specifically the inverse of the tangent function. It is used to find the angle whose tangent is a given number. In contrast to the regular tangent function, which can take any angle and return a ratio, arctan takes a ratio and returns an angle.

This function is defined as:

• Continuous: There are no breaks or gaps in the function.
• Increasing: As the value of its argument increases, the function's outcome also increases.
• Bounded: The output angles are limited between \(-\pi/2\) and \(\pi/2\).

In essence, the arctan function squashes an infinite range of inputs into its finite range of outputs. This is why, despite the tangent function going to infinity, the arctan remains bounded.

Behavior of Functions

Understanding the behavior of functions is crucial in calculus, especially when discussing limits. The behavior refers to how a function acts as the input values either increase towards infinity or decrease towards negative infinity, or approach a particular point.

For the arctan function, as mentioned earlier, it is important to highlight these behaviors:

• Monotonic Increasing: The function consistently rises as it moves from left to right on a graph, never dipping downward at any instance.
• Convergence to Limits: As the input, x, moves to extreme values, whether positive or negative infinity, the function settles closer to its boundary values (either \(-\pi/2\) or \(\pi/2\)).

These behavioral attributes allow mathematicians to predict precise outcomes for particular inputs, which is essential for calculating limits effectively.

Convergence of Functions

Convergence describes how a function approaches a specific value as the input grows large in magnitude. It is a central concept in limits, where the idea is to find what value a function tends to as its input becomes infinitely large or small.

For arctan, convergence is clearly seen:

• Toward Negative Infinity: As \(x \rightarrow -\infty\), arctan(x) converges to \(-\pi/2\).
• General Trend: As x becomes larger and larger, the values of the function move closer and closer to the boundary values, ultimately stabilizing without surpassing these bounds.

This convergence is why we can definitively say arctan(x) approaches \(-\pi/2\) as its input trends negative infinity. Unlike many other functions that might spiral or swing as infinity approaches, arctan finds a peaceful limit.