Question

Theorem 2.4 a Given the polynomial $$p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\dots+b_{1} x+b_{0}$$ prove that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\).

Short answer

Question: Prove that the limit of a polynomial function p(x) as x approaches any value 'a' is equal to the value of p(a). Answer: To prove that the limit of a polynomial function p(x) as x approaches any value 'a' is equal to p(a), we found the limit of each term in the polynomial as x approaches 'a' and then used the properties of limits to find the limit of the polynomial function. We demonstrated that the limit of p(x) and the value of p(a) are the same. Therefore, ∃ lim_{x → a} p(x) = p(a) for any value of 'a'.

Step-by-step solution

Find the limit of each term in the polynomial

For each term \(b_ix^i\) in the polynomial where \(i = 0, 1, 2, ... , n\), we need to find the limit as x approaches 'a'. The limit of a constant term (b_i) is just the constant itself, and the limit of \(x^i\) as x approaches 'a' is \(a^i\). So, the limit of each term in the polynomial function as x approaches 'a' is: $$\lim_{x \rightarrow a} b_ix^i = b_ia^i$$ We can do this for every term in the polynomial function.

Use the properties of limits to find the limit of the polynomial function

The sum of the limits of functions as x approaches 'a' is equal to the sum of the limits of each function as x approaches 'a'. Therefore, the limit of the polynomial function p(x) as x approaches 'a' is the sum of the limits of each term in the polynomial as x approaches 'a'. So, $$\lim_{x \rightarrow a} p(x) = \lim_{x \rightarrow a} \left( b_nx^n + b_{n-1}x^{n-1} + \dots + b_1x + b_0 \right)$$ Using the results from Step 1, we can rewrite this as: $$\lim_{x \rightarrow a} p(x) = b_na^n + b_{n-1}a^{n-1} + \dots + b_1a + b_0$$

Compare the limit to the value of p(a)

Now, we just need to calculate the value of p(a) and compare it to the limit we found in Step 2. The value of p(a) is: $$p(a) = b_na^n + b_{n-1}a^{n-1} + \dots + b_1a + b_0$$ We can see that this is the same as the expression we found for the limit of p(x) as x approaches 'a'.

Conclusion

Since the limit of the polynomial function p(x) as x approaches 'a' and the value of p(a) are the same, we have proven that \(\lim_{x \rightarrow a} p(x) = p(a)\) for any value of 'a'.

Key concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of variables, each multiplied by a coefficient. These coefficients are real numbers, and the powers, or exponents, are non-negative integers. The general form of a polynomial is given by: \[ p(x) = b_n x^n + b_{n-1} x^{n-1} + \cdots + b_1 x + b_0 \] where \( b_n, b_{n-1}, \ldots, b_1, b_0 \) are coefficients, and \( n \) is the degree of the polynomial. Here, \( b_n eq 0 \) (except when the polynomial is zero). The highest power of \( x \) with a non-zero coefficient determines the degree of the polynomial. Key features of polynomial functions:

• They are continuous over all real numbers, meaning you can draw them on a graph without lifting your pencil.
• The graphs of polynomials are smooth and have no sharp corners or cusps.
• End behavior is determined by the leading term \( b_n x^n \).

Limits and Continuity

A limit is a fundamental concept in calculus used to describe the behavior of a function as its input approaches a particular point. Limits help us understand how functions behave near a particular input, even if they don't actually reach that input point. In formal notation, the limit of a function \( f(x) \) as \( x \) approaches \( a \) is written: \[ \lim_{x \to a} f(x) \] Continuity is related to limits and describes a function that is unbroken or uninterrupted for its entire domain. A function \( f(x) \) is continuous at a point \( x = a \) if the following three conditions hold: - \( f(a) \) is defined. - The limit \( \lim_{x \to a} f(x) \) exists. - \( \lim_{x \to a} f(x) = f(a) \). Polynomials are continuous everywhere, which makes them particularly easy to work with when it comes to limits. Since they are defined for all real numbers and have smooth transitions, calculating their limits as \( x \to a \) simply involves evaluating the polynomial at \( a \).

Properties of Limits

When working with limits, there are several properties and rules that make them easier to compute. These properties allow us to break down complicated limits into simpler parts. Some important properties of limits include: - **Sum Rule**: The limit of a sum is the sum of the limits. If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist, then: \[ \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \] - **Product Rule**: The limit of a product is the product of the limits. Using the functions from above: \[ \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \] - **Constant Multiple Rule**: The limit of a constant times a function is the constant times the limit of the function. If \( c \) is a constant: \[ \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) \] - **Power Rule**: The limit of a function raised to a power is the limit of the function raised to that power, as long as the power itself is a positive integer. These rules apply generally, but in the case of polynomials, the process is significantly simplified. Each term in a polynomial can be handled individually using these properties and then summed to obtain the limit of the entire polynomial function. This means that to find \( \lim_{x \to a} p(x) \), we evaluate each term of the polynomial and then add them, simplifying the whole process.