Problem 53 - Calculus Early Transcendentals

Question

If p is a polynomial, show that lim \(_{x \rightarrow a} p(x)=p(a).\)

Step-by-Step Solution

Answer

The limit of a polynomial as \( x \to a \) is simply \( p(a) \), the polynomial evaluated at \( x = a \).

Step 1: Understand the Limit Definition of a Polynomial

A polynomial \( p(x) \) is a function of the form \( p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_1, a_0 \) are constants. We want to show that the limit of this polynomial as \( x \rightarrow a \) is equal to \( p(a) \). This means we substitute \( a \) into the polynomial and see that it equals the result of the limit.

Step 2: Apply Limit Properties

We use the property that the limit of a sum of functions is the sum of their limits, and the limit of a constant multiplied by a function is the constant multiplied by the limit of the function. Hence, \( \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) \).

Step 3: Use the Limit of Power

For each term \( a_i x^i \) in the polynomial, \( \lim_{x \to a} a_i x^i = a_i \cdot \lim_{x \to a} x^i = a_i a^i \), because \( \lim_{x \to a} x^i = a^i \).

Step 4: Sum the Limits of Each Term

By step 2, \( \lim_{x \to a} p(x) = \lim_{x \to a} (a_n x^n + ... + a_1 x + a_0) = a_n a^n + ... + a_1 a + a_0 \). This is equivalent to \( p(a) \).

Step 5: Conclude the Equality of Limit and Evaluation

Thus, the limit as \( x \rightarrow a \) of the polynomial \( p(x) \) simplifies to directly evaluating the polynomial at \( x = a \). Therefore, \( \lim_{x \to a} p(x) = p(a) \).

Key Concepts

Limit Properties

When working with limits in calculus, there are standard properties that greatly simplify evaluations. Two fundamental properties are useful here:

• Limit of a Sum: The limit of a sum of functions is the sum of their limits. This means if you have functions \( f(x) \) and \( g(x) \), then \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• Limit of a Constant Multiple: The limit of a constant multiplied by a function is the constant multiplied by the limit of the function. Mathematically, \( \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) \).

These properties are particularly important for handling polynomials, as they allow you to break down complex expressions into manageable parts, making the calculation of limits straightforward.

Polynomial Evaluation

Evaluating a polynomial means substituting a specific value into the polynomial expression. For a polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), to evaluate it at \( x = a \), simply replace all instances of \( x \) with \( a \).

• Example: If \( p(x) = 2x^2 + 3x + 1 \) and you want \( p(2) \), substitute 2 for \( x \) to get \( 2(2)^2 + 3(2) + 1 = 11 \).

This process confirms the polynomial's behavior at that specific point, and it directly relates to how limits operate on polynomials by ultimately finding \( p(a) \) using similar substitution principles.

Limit of a Function

The limit of a function at a particular point helps us understand the behavior of the function as it approaches that point. Specifically, for a function \( f(x) \), \( \lim_{x \to a} f(x) \) denotes the value that \( f(x) \) gets closer to as \( x \) approaches \( a \).

• For a polynomial, this means that as \( x \) approaches \( a \), the polynomial \( p(x) \) approaches the value \( p(a) \).

This connection between limits and evaluation is critical because it shows that for polynomials, limits simplify to just plugging the point into the expression — a clear, powerful attribute of polynomial functions.

Limit of a Power

Polynomials contain terms that are powers of their variable, such as \( x^n \). Understanding the limit of a power is crucial, as each term in a polynomial can be evaluated individually.

• For any base \( x \) as it approaches \( a \), \( \lim_{x \to a} x^n = a^n \).

This property ensures that when taking the limit of each term \( a_i x^i \) of the polynomial, it simplifies to \( a_i a^i \).
By applying this property across all terms, we can sum them to evaluate the limit of the entire polynomial, confirming that \( \lim_{x \to a} p(x) = p(a) \).
This step-by-step application of limits to powers within polynomials allows for straightforward limit calculations, making polynomial limit evaluations a breeze.