Problem 131 - Precalculus with Limits

Question

In Exercises 129 - 132, fill in the blanks. (Note:The notation \( x \rightarrow c^+ \) indicates that \( x \) approaches \( c \) from the right and \( x \rightarrow c^- \) indicates that \( x \) approaches \( c \) from the left.) As \( x \rightarrow \dfrac{\pi^-}{2} \), \( \tan x \rightarrow \square \) and \( \cot x \rightarrow \square \).

Step-by-Step Solution

Answer

As \( x \rightarrow \frac{\pi^-}{2} \), \( \tan x \rightarrow +\infty \) (infinity) and \( \cot x \rightarrow 0 \).

Step 1: Determine the limit of \( \tan x \) as \( x \rightarrow \frac{\pi^-}{2} \)

In the trigonometric function, the value of \( \tan x \) tends to infinity as \( x \) approaches \( \frac{\pi}{2} \) from the left side. This is because the tangent of \( \frac{\pi}{2} \) is undefined since its denominator, cos(x), becomes zero. As we approach \( \frac{\pi}{2} \) from the left, the value of \( \tan x \) increases without limit.

Step 2: Determine the limit of \( \cot x \) as \( x \rightarrow \frac{\pi^-}{2} \)

Next, consider the behavior of \( \cot x \) as \( x \) approaches \( \frac{\pi}{2} \) from the left. The cotangent of \( x \) is the reciprocal of the tangent of \( x \). As we approach \( \frac{\pi}{2} \) from the left, \( \cot x \) is actually approaching zero since \( \tan x \) is going to infinity.

Key Concepts

The Tangent Function

The tangent function, represented as \( \tan x \), is a fundamental trigonometric function. It's defined for an angle \( x \) and can be expressed as the ratio of the opposite side to the adjacent side in a right triangle. Mathematically, it's given by \( \tan x = \frac{\sin x}{\cos x} \).

However, the function exhibits special behavior around certain angles. For example, at \( x = \frac{\pi}{2} \), the cosine of \( x \), which is the denominator, becomes zero. This makes the tangent function undefined at this point. As a result, when \( x \) approaches \( \frac{\pi}{2} \) from the left, the value of \( \tan x \) tends toward infinity.

• The tangent function is periodic with a period of \( \pi \).
• Its values range from negative infinity to positive infinity.
• The function has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).

Understanding this behavior is crucial when evaluating limits involving the tangent function.

Approaching Limits

When working with functions, especially trigonometric ones, understanding limits is pivotal. Limits define the value that a function approaches as the input approaches a particular point. Specifically, in this exercise, the focus is on approaching \( x \rightarrow \frac{\pi^-}{2} \), which means approaching \( \frac{\pi}{2} \) from the left side.

To comprehend this concept, consider the following points:

• As \( x \) inch towards \( \frac{\pi}{2} \) from the left, \( \tan x \) tends to become infinitely large because \( \cos x \) approaches zero, making the ratio \( \frac{\sin x}{\cos x} \) very large.
• Conversely, as \( x \) approaches \( \frac{\pi}{2} \) from the left, \( \cot x \) approaches zero. This is because \( \cot x = \frac{1}{\tan x} \), so as \( \tan x \) increases significantly, \( \cot x \) becomes smaller and smaller.

Approaching limits enables us to grasp the behavior of functions near points where they are not explicitly defined.

Trigonometric Limits

Trigonometric limits involve evaluating the behavior of trigonometric functions as their arguments approach specific values. These limits are essential in calculus, particularly when dealing with infinite limits or limits approaching undefined points.

For trigonometric limits such as \( \tan x \) at \( \frac{\pi^-}{2} \):

• The limit is evaluated by considering the trigonometric function's behavior as it nears an undefined value.
• Similarly, for \( \cot x \) near \( \frac{\pi}{2} \), it approaches zero due to its reciprocal nature with tangent.

These limits showcase how functions behave as they approach critical points, especially when these points cause a division by zero scenario. Calculating such limits provides insights into the trend a function follows as it nears these undefined or infinite points, an essential skill in mathematical analysis.