Problem 67 - Calculus 3

Question

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(\pi / 4,1)} \frac{y \tan x}{y+1} $$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{2}\).

Step 1: Substitute the limit values into the expression

Begin by substituting the values \(x = \frac{\pi}{4}\) and \(y = 1\) into the expression \(\frac{y \tan x}{y+1}\). This gives us \(\frac{1 \cdot \tan \frac{\pi}{4}}{1+1}\).

Step 2: Evaluate \(\tan \frac{\pi}{4}\)

Remember that \(\tan \frac{\pi}{4} = 1\), since it is a standard trigonometric value. Therefore, substitute this into our expression to have \(\frac{1 \cdot 1}{1+1}\).

Step 3: Simplify the expression

Simplify the expression \(\frac{1}{2}\). This is the result after substitution and simplification.

Key Concepts

Multivariable Functions

In multivariable calculus, we deal with functions of two or more variables. Multivariable functions extend the concepts of single-variable functions to multiple dimensions, which allows us to describe more complex phenomena. For example, in the given problem, we have a function of two variables, \(x\) and \(y\), which is expressed as \(\frac{y \tan x}{y+1}\). Here are a few key points to better understand multivariable functions:

• Inputs: The variables \(x\) and \(y\) serve as inputs to the function; their values determine the output.
• Graphical Representation: In a three-dimensional space, the graph of a function \(f(x, y)\) is a surface.
• Domain: The domain of a multivariable function is a set of points \((x, y)\) for which the function is defined.

Understanding these concepts helps in approaching problems like finding limits where the behavior at specific points in the domain is of interest.

Trigonometric Limits

Trigonometric limits play a crucial role in calculus, especially when dealing with periodic functions such as sine, cosine, and tangent. In our problem, evaluating \( \tan \frac{\pi}{4} \) is essential since this is the trigonometric part of the expression. Here are some important aspects of trigonometric limits:

• Standard Values: Familiarity with standard trigonometric values, such as \( \tan \frac{\pi}{4} = 1 \), \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), and others, speeds up calculations.
• Continuity: Most basic trig functions are continuous, meaning their limits evaluate to the function's value at that point if the function is defined at that point.
• Identity Usage: Trigonometric identities, such as \( \tan x = \frac{\sin x}{\cos x} \), can help in simplifying expressions before evaluating limits.

These properties make it easier to evaluate limits in complex expressions that involve trig functions.

Limit Existence

Understanding whether a limit exists is fundamental in calculus. The concept of limit existence, especially in multivariable functions, is slightly more complex than in single-variable calculus.In the exercise, our objective was to evaluate the limit of a function as \( (x, y) \) approaches \( (\frac{\pi}{4}, 1) \). Here's a step-by-step approach to grasp the concept:

• Substitute Values: In multivariable limits, we first substitute approaching values directly into the function to check if a consistent value is reached.
• Consistency: If the function approaches different values along different paths to the point, the limit does not exist.
• Simplification: Simplifying the function can sometimes reveal whether the limit exists more clearly.

For our problem, substituting and simplifying showed that the limit exists, with a consistent value of \( \frac{1}{2} \). Recognizing when and how a limit exists is crucial for solving calculus problems effectively.