Problem 70 - Precalculus

Question

Given \(f(x)=\cos (\tan x),\) find $$\lim _{x \rightarrow \frac{\pi}{2}} f(x)$$

Step-by-Step Solution

Answer

The limit does not exist because \(\cos(\tan x)\) oscillates between -1 and 1.

Step 1: Understanding the Function

We are given the function \( f(x) = \cos(\tan x) \). We need to find the limit as \( x \) approaches \( \frac{\pi}{2} \). The function comprises two parts: the tangent function and the cosine function. As we approach \( x = \frac{\pi}{2} \), we must understand the behavior of \( \tan x \) and \( \cos \) for those values of \( x \).

Step 2: Evaluate the Behavior of \(\tan x\)

The tangent function, \( \tan x \), has a vertical asymptote at \( x = \frac{\pi}{2} \). As \( x \) approaches \( \frac{\pi}{2} \) from the left, \( \tan x \) increases towards positive infinity. From the right, \( \tan x \) decreases towards negative infinity.

Step 3: Consider the Properties of \(\cos x\)

The cosine function, \( \cos(\theta) \), is defined and oscillates between -1 and 1 for all real \( \theta \). No matter how large or how negative \( \theta \) becomes, \( \cos(\theta) \) will always be within the interval [-1, 1].

Step 4: Approach \(\lim_{x \to \frac{\pi}{2}} \cos(\tan x)\)

Given the nature of \( \tan x \) as \( x \to \frac{\pi}{2} \), \( \tan x \) tends to \( \pm \infty \). However, \( \cos(\theta) \) does not approach a limit as \( \theta \) goes to infinity or negative infinity, but rather oscillates between -1 and 1. This means that \( \cos(\tan x) \) does not settle at any specific value as \( x \to \frac{\pi}{2} \).

Step 5: Conclusion on the Limit

Since \( \cos(\tan x) \) does not converge to any specific value but continues to oscillate between -1 and 1 as \( x \to \frac{\pi}{2} \), the limit \( \lim_{x \to \frac{\pi}{2}} f(x) \) does not exist.

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is key to solving limits involving them. In this exercise, we work with two trigonometric functions: tangent and cosine.

• Tangent Function (\(\tan x\)): This function represents the ratio of the sine to the cosine of an angle. As you approach special points like \(\frac{\pi}{2}\), the function's behavior becomes unique with significant changes.
• Cosine Function (\(\cos x\)): This function measures the horizontal length of a point on the unit circle moving through an angle. It's periodic, with values that repeat every \(2\pi\). The cosine value is always between -1 and 1, showing a regular, predictable pattern, unlike tangent.

The combination of these functions in a limit problem can create challenging situations to analyze, such as infinite oscillation.

Asymptotes

An asymptote refers to a line that a graph approaches but never actually reaches as the input values become extreme.
Tangent functions have vertical asymptotes where the function grows towards positive or negative infinity. Specifically, \(\tan x\) has a vertical asymptote at each odd multiple of \(\frac{\pi}{2}\), such as at \(x = \frac{\pi}{2}\). This means as \(x\) approaches \(\frac{\pi}{2}\) from either side, \(\tan x\) becomes unbounded:

• Approaches positive infinity from the left.
• Approaches negative infinity from the right.

These extreme values then impact trigonometric limits significantly, complicating the behavior of the composite function \(\cos(\tan x)\).

Function Behavior

The behavior of a function helps determine its limits as the input approaches certain values. When analyzing \(f(x) = \cos(\tan x)\), understanding the interplay between tangent and cosine is essential.

• Tangent's Extreme Behavior: As you approach \(x = \frac{\pi}{2}\), \(\tan x\) exhibits very rapid changes, going from negative infinity to positive infinity.
• Cosine's Oscillation: Although \(\tan x\) assumes extreme values, \(\cos(\tan x)\) simply oscillates between -1 and 1 without settling. It doesn't follow a straightforward growth or decay pattern.

This combination results in a function without a clear direction or value at \(x = \frac{\pi}{2}\), leading the limit not to exist there.

Infinite Oscillation

Infinite oscillation indicates a function fluctuates continuously without approaching a finite limit.
For \(f(x) = \cos(\tan x)\), as \(x\) approaches \(\frac{\pi}{2}\), \(\tan x\) shoots to infinite positive and negative values. Consequently, \(\cos(\tan x)\) endlessly cycles through its range of -1 to 1.

• This persistent fluctuation between -1 and 1 creates an indefinite pattern.
• Without stabilization towards any single value, the limit is non-existent.

Understanding infinite oscillation is vital for grasping why some limits, like \(\lim_{x \rightarrow \frac{\pi}{2}} \cos(\tan x)\), don't exist, despite having the appearance of predictability.