Problem 57 - Calculus: Early Transcendentals

Question

If \( p \) is a polynomial, show that \( \displaystyle \lim_{x \to a}p(x) = p(a) \).

Step-by-Step Solution

Answer

The limit of a polynomial \( p(x) \) as \( x \to a \) equals \( p(a) \).

Step 1: Understanding Polynomials

A polynomial function is a function of the form \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_0, a_1, \ldots, a_n \) are constants, and \( n \) is a non-negative integer. The limit of the polynomial as \( x \to a \) needs to be evaluated.

Step 2: Applying the Limit Definition to Each Term

The limit of a sum of functions is the sum of their limits. Thus, the limit \( \lim_{x \to a}p(x) \) is \( \lim_{x \to a}(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0) \). This can be broken down into \( a_n \lim_{x \to a}x^n + a_{n-1} \lim_{x \to a}x^{n-1} + \ldots + a_1 \lim_{x \to a}x + a_0 \lim_{x \to a}1 \).

Step 3: Calculating Limits of Individual Terms

For each term, use the property \( \lim_{x \to a}x^k = a^k \). This gives us \( a_n a^n + a_{n-1} a^{n-1} + \ldots + a_1 a + a_0 \).

Step 4: Combining the Limits

Combine the results from Step 3. Since the limit of each term matches its value when \( x = a \), this results in \( p(a) \). More explicitly, the polynomial evaluated at \( a \) is \( a_n a^n + a_{n-1} a^{n-1} + \ldots + a_1 a + a_0 \), which is precisely \( p(a) \).

Step 5: Concluding the Limit Proof

Since for any polynomial \( p(x) \), \( \lim_{x \to a}p(x) = p(a) \) has been shown by evaluating each term, this proves that the limit of a polynomial as \( x \to a \) is precisely equal to the polynomial evaluated at \( a \).

Key Concepts

Polynomials

Polynomials are fundamental mathematical expressions involving variables raised to different powers with coefficients. In general, they have the form:

• \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Where \( a_0, a_1, \ldots, a_n \) are constants, which determine the specific characteristics of the polynomial. The variable \( x \) is raised to various powers, and \( n \) represents the highest power, which also dictates the polynomial's degree. This simple yet versatile structure allows polynomials to model many real-world situations. Understanding their forms and properties helps us solve various mathematical problems efficiently with clarity.

Limit Laws

In calculus, limit laws are rules that help us evaluate the limits of functions as a variable approaches a particular value. They allow us to break down complex expressions into simpler parts. Some essential limit laws include:

• **Sum Law**: \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
• **Product Law**: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \)
• **Power Law**: \( \lim_{x \to a} [f(x)]^n = (\lim_{x \to a} f(x))^n \)

These laws simplify the process of evaluating the limits of polynomial functions by allowing each polynomial term to be handled independently. Hence, when evaluating \( \lim_{x \to a} p(x) \), we apply these laws to evaluate each term in the polynomial separately, making the problem more manageable.

Continuity

A function is said to be continuous when there is no interruption in its graph. This means that as you approach any point in the function, the function smoothly continues through that point without any gaps, jumps, or holes. For polynomials, continuity is a natural property. Because polynomials lack undefined points, vertical asymptotes, or sudden jumps, they are continuous everywhere on their domains. Consequently, evaluating the limit of a polynomial as \( x \) approaches a particular value is straightforward; the value of the polynomial at that point is simply \( p(a) \). Thus, polynomials are excellent candidates for demonstrating continuity and the ease of limit evaluation.

Evaluation of Limits

To evaluate the limit of a polynomial function as \( x \) approaches a specific value \( a \), we use the continuity property of polynomials. Given the polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), the limit as \( x \to a \) can be determined by direct substitution:

• Replace \( x \) with \( a \) in each term of the polynomial.
• Calculate each term: \( a_n a^n, a_{n-1} a^{n-1}, \ldots, a_1 a, \) and finally \( a_0 \).
• Add the results together to find \( p(a) \).

This approach works seamlessly for polynomials due to their continuous nature, confirming that \( \lim_{x \to a} p(x) = p(a) \). It highlights the simplicity of evaluating limits for polynomials, as direct evaluation is not only valid but proves definitively the value of the limit.