Problem 67 - Calculus Early Transcendentals

Question

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the function as (x, y) approaches (2, 0): $\lim_{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}$. Answer: $\frac{1}{4}$.

Step 1: Identify the Known Limit Formula

We need to find the limit: $$ \lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}.$$ Let's first analyze the numerator. We can use the following well-known limit related to the cosine function: $$\lim _{u \rightarrow 0} \frac{1-\cos u}{u^{2}} = \frac{1}{2}.$$

Step 2: Apply the Known Limit Formula

To use the known limit to evaluate the given limit, we will perform a substitution. Let \(u = y\) and rewrite the given limit as follows: $$\lim _{(x, u) \rightarrow(2, 0)} \frac{1-\cos u}{x u^{2}}.$$ Applying the known limit, we have: $$\lim _{(x, u) \rightarrow(2, 0)} \frac{1-\cos u}{x u^{2}} = \frac{1}{2}\lim _{(x, u) \rightarrow(2, 0)} \frac{1}{x}.$$

Step 3: Evaluate the Remaining Limit

Now, we only need to evaluate the following limit: $$\frac{1}{2}\lim _{(x, u) \rightarrow(2, 0)} \frac{1}{x}.$$ Since the limit depends only on \(x\), we can ignore \(u\): $$\frac{1}{2}\lim _{x \rightarrow 2} \frac{1}{x}.$$ Plugging in \(x=2\), we get: $$\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}.$$ Therefore, the limit of the given function is: $$\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}} = \frac{1}{4}.$$

Key Concepts

Cosine function

The cosine function is a fundamental part of trigonometry and is often involved in calculus problems. A special property of the cosine function, particularly near zero, is that it can be approximated using its Taylor series expansion. This behavior is useful for evaluating limits involving cosine. In calculus, especially for limits approaching zero, we utilize the fact that the cosine function can be represented as:
\[1 - \cos u \approx \frac{u^2}{2}\]This approximation leads to the well-known limit \( \lim_{u \to 0} \frac{1 - \cos u}{u^2} = \frac{1}{2} \).

• This approximation is pivotal when dealing with limits involving terms like \( 1 - \cos u \) in the numerator as \( u \to 0 \).
• When substituting variables, keeping this approximation in mind simplifies complex expressions, improving our ability to calculate multivariable limits.

Multivariable limits

Multivariable limits involve evaluating the behavior of a function as more than one variable approaches a particular point. Unlike single-variable limits, multivariable limits consider the changes of all variables simultaneously.

• When solving for a multivariable limit, such as \( \lim_{(x, y) \to (2, 0)} \frac{1-\cos y}{x y^{2}} \), both variables head towards specific values.
• Analyzing paths, such as letting \( y \) approach zero while maintaining the condition on \( x \), helps us ensure that the limit is consistent across different approaches.
• Multivariable limits can be quite complex, but important techniques include changing the variables to simplify the problem and focusing on critical points of interest.

For instance, using substitution techniques to temporarily simplify the two variables to a more manageable form—such as "Setting \( u = y \)"—helps streamline the equation to resemble more straightforward single-variable limits.

Limit substitution

Limit substitution is a calculus technique used to simplify the evaluation of limits by substituting variables or expressions. This method helps by transforming the problem into a more recognizable form.
In the context of the given expression:\[\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}\]we applied substitution by setting \( u = y \). This allows us to leverage the known limit:\[ \lim _{u \rightarrow 0} \frac{1-\cos u}{u^{2}} = \frac{1}{2} \].
While solving limits:

• Substitution can make complex multivariable expressions easier to handle.
• Identifying patterns or known limits after substitution is crucial.
• Once the problematic variable is isolated, compute the limit as you would with simpler terms.

Ultimately, substitution is a powerful tool in calculus that can drastically reduce the complexity of problems, turning a seemingly daunting multivariable situation into a manageable single-variable solution.