Questions: Compare your answers for parts (a) and (b). What powers of y in the numerator do you think would behave more like the one in (a)? what powers of y in the numerator do you think would behave more like part (b)? Why?

Compare your answers for parts (a) and (b). What powers of y in the numerator do you think would behave more like the one in (a)? what powers of y in the numerator do you think would behave more like part (b)? Why?

Transcript text: Compare your answers for parts (a) and (b). What powers of $y$ in the numerator do you think would behave more like the one in (a)? what powers of $y$ in the numerator do you think would behave more like part (b)? Why?

Solution

Solution Steps

To solve the given problem, we need to analyze the behavior of the given expressions as $x$ and $y$ approach certain limits. We will focus on the first three parts of the question:

For the expression $\frac{y^4}{x^4 + y_1}$, we need to understand how the power of $y$ in the numerator affects the behavior of the fraction as $x$ and $y$ change.
For the expression $\lim_{(x, y) \to (0,0)} \frac{x y^3}{x^4 + y''}$, we will compute the limit to see how the expression behaves as both $x$ and $y$ approach zero.
For the expression $\frac{x y^4}{x^n + y^4}$, we will analyze how different powers of $x$ in the denominator affect the behavior of the fraction.

Step 1: Analyze Expression 1

The first expression is given by

\[ \frac{y^4}{x^4 + y_1} \]

To understand its behavior, we note that as $y$ increases, the term $y^4$ in the numerator will dominate the expression if $x^4$ remains relatively small compared to $y_1$. Thus, the behavior of this expression is heavily influenced by the power of $y$ in the numerator.

Step 2: Evaluate Limit of Expression 2

The second expression is

\[ \lim_{(x, y) \to (0,0)} \frac{x y^3}{x^4 + y''} \]

As both $x$ and $y$ approach zero, the numerator $x y^3$ also approaches zero. The denominator $x^4 + y''$ approaches $y''$ (assuming $y''$ is not zero). Therefore, the limit evaluates to:

\[ \lim_{(x, y) \to (0,0)} \frac{x y^3}{y''} = 0 \]

Step 3: Analyze Expression 3

The third expression is

\[ \frac{x y^4}{x^n + y^4} \]

In this case, the behavior of the expression depends on the relative sizes of $x^n$ and $y^4$. If $y$ is large compared to $x$, then $y^4$ will dominate, and the expression will behave like

\[ \frac{x y^4}{y^4} = x \]

Conversely, if $x$ is large compared to $y$, the expression will behave like

\[ \frac{x y^4}{x^n} = \frac{y^4}{x^{n-1}} \]

Final Answer

The analysis shows that:

The expression $\frac{y^4}{x^4 + y_1}$ behaves significantly based on the power of $y$ in the numerator.
The limit of $\frac{x y^3}{x^4 + y''}$ as $(x, y) \to (0,0)$ is $0$.
The expression $\frac{x y^4}{x^n + y^4}$ behaves differently based on the relative sizes of $x$ and $y$.

Thus, the final answers are:

\[ \boxed{0} \] for the limit, and the behavior of the other expressions is dependent on the context provided.

Was this solution helpful?