91Ó°ÊÓ

A polynomial function is a mathematical expression consisting of variables, coefficients, and exponents, which are combined using addition, subtraction, multiplication, and non-negative integer exponents. For example, the polynomial function mentioned in the exercise is \(4x^2 + 1\), which includes the variable \(x\), raised to the second power, and coefficients 4 and 1.

One of the characteristics of these functions is their smooth and continuous nature, which means they don’t have breaks, holes, or sharp corners in their graphs. Polynomial functions are also differentiable and integrable. A key aspect of polynomial functions, which plays a critical role in limit evaluation, is that they are identical to their limit at any point within their domain. That means the limit as \(x\) approaches a particular value is equal to the function's value at that point, provided that the point is within the function's domain.

Understanding the limit properties is essential in evaluating limits in calculus. Limit properties are mathematical rules that allow for the simplification and calculation of limits of functions. They are based on the behavior of functions as the input approaches a certain value but does not necessarily reach it.

Some of the key limit properties include:

• The limit of a constant is the constant itself.
• The limit of a function multiplied by a constant is the constant times the limit of the function.
• The limit of a sum or difference is the sum or difference of the limits.
• If two functions are equal near a point, their limits as they approach that point are also equal.

Applying these properties can drastically simplify the process of finding limits, especially with polynomial functions. As seen in the step-by-step solution, the property that the limit of a polynomial function as \(x\) approaches a value is just the value of the function when \(x\) is replaced by that number was used to find the limit.

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It consists of a series of logically deduced steps that derive a conclusion from a set of premises, which include definitions, axioms, previously proven theorem, and given statements. In the context of calculus and limits, proofs are used to verify whether the limit of a function as \(x\) approaches a certain value indeed equals a specific number.

In our exercise, the proof required showing that \( \lim _{x \rightarrow 1}(4x^2 + 1) = 5 \). The step-by-step solution provided in the original exercise used direct substitution, which is one of the simplest and most powerful tools for solving limits of polynomial functions. This method is based on the premise that a polynomial function is continuous, and evaluating the function at a certain point gives the same result as taking the limit at that point. The proof concludes with the verification that the value obtained by substituting \(x = 1\) into the function equals 5, thereby confirming the limit.