limit x tends to infinity y tends to three 2x-3 / x^3 + 4y^3

To find the limit of the given expression:

\[
\lim_{{x \to \infty, y \to 3}} \frac{2x - 3}{x^3 + 4y^3}
\]

We need to analyze the behavior of the function as \(x\) tends to infinity and \(y\) tends to 3.

### Step-by-Step Solution:

The given expression is:

\[
\frac{2x - 3}{x^3 + 4y^3}
\]

As \(y \to 3\), we can substitute \(y = 3\) in the denominator to simplify the expression:

\[
4y^3 = 4(3)^3 = 4 \times 27 = 108
\]

So, the expression becomes:

\[
\frac{2x - 3}{x^3 + 108}
\]

### Behavior as \(x \to \infty\):

- The numerator \(2x - 3\) grows linearly with \(x\), meaning it approaches infinity as \(x\) increases.
- The denominator \(x^3 + 108\) grows cubically with \(x\), and as \(x \to \infty\), the term \(x^3\) dominates, making \(108\) negligible.

Thus, as \(x\) becomes very large, the function behaves approximately like:

\[
\frac{2x}{x^3} = \frac{2}{x^2}
\]

As \(x \to \infty\), \(\frac{2}{x^2} \to 0\).

### Final Answer:

\[
\lim_{{x \to \infty, y \to 3}} \frac{2x - 3}{x^3 + 4y^3} = 0
\]

Would you like more details on any part of this solution? Here are some related questions to explore:

1. What happens to higher-order terms when computing limits as \(x \to \infty\)?
2. How do limits behave when both variables approach infinity?
3. Can we use L'Hopital's Rule to solve this limit?
4. What is the importance of the dominant term in limits?
5. How do limits help in understanding asymptotic behavior of functions?

**Tip:** When dealing with limits as \(x \to \infty\), focus on the highest degree terms in both the numerator and the denominator to simplify the calculation.

This page explains how to find the limit of the expression (2x - 3) / (x^3 + 4y^3) as x tends to infinity and y tends to 3. Using key limit theorems and analyzing the dominant terms, the solution walks through each step to show why the final result is 0. Ideal for students in Grades 11-12 studying limits and asymptotic behavior.

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