91Ó°ÊÓ

Polynomial functions are mathematical expressions that involve variables raised to whole number powers. They are presented in the form:

• Constant terms: A simple number standing on its own.
• Variables: Letters that represent numbers.
• Exponents: Powers to which the variables are raised.
• Coefficients: Numbers preceding the variables or constants.

The general form for a polynomial function is given by:
\[f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\]
where \(n\) is a non-negative integer and \(a_n, a_{n-1}, ..., a_1, a_0\) are constants called the coefficients. For example, in the polynomial \(f(x) = x^2 - 6x + 10\), \(x^2\) is a term with a power of 2, \(-6x\) is a term with a power of 1, and 10 is a constant term.
Polynomial functions have a very straightforward rule of operations and are some of the simplest functions to work with. This simplicity often allows us to use the Direct Substitution Method easily.

Continuity in functions is a key concept in calculus. A function is continuous at a point \(x = a\) if there are no interruptions, jumps, or breaks in the graph at that point. Specifically, for a function \(f(x)\) to be continuous at \(x = a\), three conditions must be met:

• The function \(f(x)\) must be defined at \(x = a\).
• The limit \(\lim_{x \to a} f(x)\) must exist.
• The value of the function at \(x = a\), noted as \(f(a)\), must equal the limit reached as \(x\) approaches \(a\): \(f(a) = \lim_{x \to a} f(x)\).

Polynomial functions, such as \(f(x) = x^2 - 6x + 10\), are continuous everywhere on their domain, meaning they satisfy the conditions for continuity at all points. This property simplifies the process of evaluating limits, particularly when using direct substitution.

Evaluating limits is a fundamental process in calculus, used to understand the behavior of functions as inputs approach a particular value. The limit of a function \(f(x)\) as \(x\) approaches a value \(a\) is denoted as \(\lim_{x \to a}f(x)\).
The Direct Substitution Method is a straightforward way to evaluate limits when dealing with functions known to be continuous at the point of interest. For a polynomial function, where continuity is guaranteed, you simply substitute the \(x\) value directly into the function.
In our example, evaluating \(\lim_{x \to 1}(x^2 - 6x + 10)\), involves:

• Substituting \(x = 1\) into the polynomial: \(f(1) = 1^2 - 6 \cdot 1 + 10\).
• Simplifying the expression: \(1 - 6 + 10\).
• Calculating the result: 5.

This method efficiently yields the limit because the polynomial’s continuity assures us that the function value at \(x = 1\) matches the limit as \(x\) approaches 1.