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The concept of the degree of a polynomial is foundational in understanding limits, especially when dealing with rational expressions. A polynomial's degree is the highest power of the variable present in the expression. In our example, the function is a rational expression:\( f(x) = \frac{6x^2 + 5x + 100}{3x^2 - 9} \).
Here, you can identify the degree of the polynomial in both the numerator and the denominator by looking at the highest exponent of \(x\).- **Numerator Degree**: The highest power of \(x\) in the numerator \(6x^2 + 5x + 100\) is 2, due to the term \(6x^2\).- **Denominator Degree**: Similarly, in the denominator \(3x^2 - 9\), the highest power of \(x\) is also 2, due to \(3x^2\).
It's crucial to identify these degrees because they guide you in finding limits, particularly in determining the behavior as \(x\) approaches infinity.

When working with rational expressions, knowing how to handle the numerator and the denominator separately can dramatically simplify your understanding. A fraction consists of two groups: - **The Numerator**: This is the top part of the fraction. In our function, the numerator is\( 6x^2 + 5x + 100 \).- **The Denominator**: This is the bottom portion of the fraction, here \(3x^2 - 9 \).
The key to solving limits is often in balancing these two parts. By analyzing their degrees, you can determine how dividing each term by the highest power affects the expression.
Here, since the degrees are identical (both are 2), dividing each term by \(x^2\) aligns the terms in a way that allows for simplification as \(x\) approaches infinity.

An infinity limit refers to the behavior of a function as \(x\) approaches positive or negative infinity. In simple terms, it helps us understand how the entire expression behaves as \(x\) becomes very large.
For the given function:\( \lim _{x \rightarrow+\infty} \frac{6 x^{2}+5 x+100}{3 x^{2}-9} \), the goal is to figure out the function's end behavior. By dividing each term by \(x^2\), the analysis becomes clear:- As \(x \to +\infty\), terms like \(\frac{5}{x}\) and \(\frac{100}{x^2}\) in the numerator approach 0 since the variable \(x\) is in the denominator.- Similarly, \(\frac{9}{x^2}\) in the denominator also approaches zero.
Thus, the limit simplifies to \(\frac{6}{3} = 2\) because these diminishing terms no longer affect the leading terms.

Simplifying rational expressions is a critical step in finding limits. It allows you to reduce the expression to a simpler form, which is easier to evaluate as \(x\) approaches infinity. Here are the steps in the process we used:

• **Step 1**: Identify the degree of the numerator and the denominator. Both degrees are 2.
• **Step 2**: Divide each term in the expression by the highest power of \(x\) in the denominator, which is \(x^2\).
• **Step 3**: Simplify the resulting terms. Terms where \(x\) is in the denominator, like \(\frac{5}{x}\) and \(\frac{100}{x^2}\), reduce to zero as \(x\) approaches infinity.
• **Step 4**: Evaluate the simplified expression to find the limit.

By following these steps, you extract the core behavior of the function, leading to the simplified limit \(2\).