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In calculus, infinite limits refer to the behavior of a function as the input value approaches infinity. When we say \( \lim_{x \to \infty} f(x) \), we want to see what happens to \( f(x) \) as \( x \) gets really large.
In problems involving rational functions, as \( x \) goes to infinity, we often look at which terms dominate the expression. Usually, these are terms with the highest power of \( x \).
In our exercise, we simplified by dividing all terms by \( x^2 \), which helps us focus on the dominant terms. Simplification not only makes calculations easier but also highlights how the function behaves infinitely.

The cube root limit involves finding the limit of a function that includes a cube root. The cube root is a special type of root where instead of two factors that multiply to give the original value (as in square roots), you have three.
So when we encounter an expression like \( \sqrt[3]{\frac{-5}{8}} \), we are finding what happens to this expression as our variable approaches infinity. The cube root function smoothens the changes in a function's behavior, making it important in analyzing limits.
Knowing how to deal with the cube root of both positive and negative numbers is key. For example, the cube root of a negative number like \( -1 \) is another negative number \( -1 \), unlike square roots which are undefined for negatives in real numbers.

Rational functions are simply fractions where both the numerator and the denominator are polynomials. Understanding these can greatly aid in limit problems.
For the given function, \( \frac{2+3x-5x^2}{1+8x^2} \), it's clear that as \( x \to \infty \), terms with the highest power, \( -5x^2 \) and \( 8x^2 \), will determine the slope of the function. These dominate because they grow much faster than the other terms.
Simplifying rational functions helps identify these dominant terms and eases the path to calculating limits. It's crucial to simplify whenever possible so that we can clearly see which terms dictate the function's behavior as \( x \to \infty \).

When finding limits, following a set of steps can clarify the problem. Here’s what was done in our solution:

• **Simplification**: Divide every term by \( x^2 \) to simplify the expression inside the root. This highlights the most significant contributors as \( x \to \infty \).


• **Evaluate the Simplified Expression**: Recognize that terms like \( 2/x^2 \) and \( 3/x \) vanish as \( x \) increases, leaving the core of the function.


• **Compute the Limit of the Root**: Once simplified, compute the cube root, as done with \( -5/8 \). Always evaluate using simplified figures to avoid mistakes.


• **Final Calculation**: Calculate the numerical root value. Here, the cube root remains constant, \( -\sqrt[3]{\frac{5}{8}} \), confirming our result.

By following these steps diligently, you ensure that every aspect of the problem is understood and calculated correctly.