Problem 49 - Thomas Calculus

Question

Find the limits in Exercises \(49-52\). $$\lim _{x \rightarrow(\pi / 2)^{-}} \tan x$$

Step-by-Step Solution

Answer

The limit is \( +\infty \).

Step 1: Understanding the Function and Expression

We are asked to find the limit \( \lim_{x \rightarrow (\pi/2)^{-}} \tan x \). This means we need to determine the behavior of the tangent function as \( x \) approaches \( \pi/2 \) from the left (from values less than \( \pi/2 \)).

Step 2: Recall the Tangent Function's Behavior

The tangent function \( \tan x \) is defined as \( \frac{\sin x}{\cos x} \). As \( x \) approaches \( \pi/2 \) from the left, \( \sin x \) approaches 1, while \( \cos x \) approaches 0 from the positive side. Thus, the fraction approaches a form where 1 is divided by a very small positive number, indicating that \( \tan x \) grows very large positively.

Step 3: Concluding the Behavior Near the Point

Since \( \tan x \) grows without bounds positively as \( x \rightarrow (\pi/2)^{-} \), we conclude that the limit is approaching infinity. In mathematical notation, this is represented as \( +\infty \).

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry. It is defined as the ratio of the sine function to the cosine function, specifically:\[ \tan x = \frac{\sin x}{\cos x} \]This definition gives the tangent function some interesting properties, especially when the cosine of an angle approaches zero. Because you're dividing by cosine,

• if cosine is positive and exceedingly small, tangent becomes a large positive number,
• if cosine is negative and exceedingly small, tangent becomes a large negative number.

Understanding the behavior of the tangent function is crucial for solving limits involving it, especially near points where cosine is zero, such as at multiples of π/2 (90 degrees).
In these scenarios, you'll observe the tangent function approaching infinity or negative infinity, depending on the direction of the limit approach.

One-Sided Limits

One-sided limits are strategies for evaluating the behavior of functions as they approach a specific point from only one direction. This is useful when functions behave differently when approached from the left versus the right.
For example, when determining \( \lim _{x \rightarrow (\pi / 2)^{-}} \tan x \), it means:

• We are interested in the values of \( x \) that are less than \( \pi/2 \).
• As \( x \) becomes very close to \( \pi/2 \), but remains less than it, the behavior of the function is analyzed.

This specific one-sided limit shows how the tangent function behaves as it nears \( \pi/2 \) from the left. As the cosine of \( x \) approaches zero from the positive side, \( \tan x \) increases hugely towards positive infinity, demonstrating the concept of divergence towards infinity.

Trigonometric Limits

Trigonometric limits involve evaluating the limits of functions that contain trigonometric functions, such as sine, cosine, and tangent. These types of limits can often involve interesting behavior due to the periodic and continuous nature of trigonometric functions.

• For \( \tan x = \frac{\sin x}{\cos x} \), the limit heavily depends on the behavior of sine and cosine near the point of concern.
• When approaching specific points such as \( \pi/2 \), where cosine is zero, the limit of tangent can become problematic, causing undefined or infinite behavior.

Understanding trigonometric limits is crucial because they often don't behave simply like polynomial limits as you approach boundary values. Recognizing that, near boundary points where typical trigonometric values are undefined due to division by zero, allows us to predict and solve limit problems successfully, as seen with: \[ \lim _{x \rightarrow (\pi/2)^{-}} \tan x = +\infty \]