Problem 3 - Calculus Early Transcendentals

Question

What does it mean to say that limits of polynomials may be evaluated by direct substitution?

Step-by-Step Solution

Answer

How are Polynomials related? Can you explain using Direct Substitution to evaluate a polynomial's limit? Answer: A Limit is a fundamental concept in calculus. It describes the behavior of a function as its input approaches a specific value. A Polynomial is a mathematical expression made up of variables, coefficients, and exponents. If we want to find the limit value ('L') of a function (f(x)) as x approaches a certain point (a), we denote it as: $$\lim\limits_{x \to a} f(x) = L$$ Polynomials are continuous functions, meaning their graphs do not have any breaks or jumps. Because of this continuity, when we calculate the limit of a polynomial, we can use a method called Direct Substitution. Here, we simply replace 'x' with the value 'a': $$\lim\limits_{x \to a} P(x) = P(a)$$ For example, to find the limit of the polynomial \(P(x) = 3x^2 + 2x - 1\) as \(x\) approaches 2, we substitute 'x' with 2 to get: $$P(2) = 3(2)^2 + 2(2) - 1 = 15$$ So the limit of the polynomial as \(x\) approaches 2 is 15.

Step 1: Introduction to Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input (or argument) approaches a certain value. The limit of a function f(x) as x approaches a certain value 'a' can be denoted as: $$\lim\limits_{x \to a} f(x)$$ If the function approaches a specific value as x approaches 'a', then it is said to have a limit, and we write: $$\lim\limits_{x \to a} f(x) = L$$ where L is the limit value of the function.

Step 2: Introduction to Polynomials

A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable raised to a power and multiplied by a coefficient. A polynomial with one variable (x) can be written as: $$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where \(a_n, a_{n-1}, \cdots, a_1, a_0\) are the coefficients and 'n' is a non-negative integer.

Step 3: Direct Substitution in Polynomials

When calculating the limit of polynomials, we can use a method called direct substitution. It means that we can simply substitute the value of 'a' (the point we want to find the limit) into the function: $$\lim\limits_{x \to a} P(x) = P(a)$$ This method works for polynomials because they are continuous functions, meaning that their graphs do not have any breaks or 'jumps' as the input values change.

Step 4: Evaluating a Limit Using Direct Substitution

As an example, let's find the limit of the polynomial \(P(x) = 3x^2 + 2x - 1\) as \(x\) approaches 2: $$\lim\limits_{x \to 2} (3x^2 + 2x - 1)$$ Using direct substitution, we can replace 'x' with the value 2: $$P(2) = 3(2)^2 + 2(2) - 1$$ Now, calculate the result: $$P(2) = 3(4) + 4 - 1 = 12 + 4 - 1 = 15$$ Hence, the limit of the polynomial as \(x\) approaches 2 is 15: $$\lim\limits_{x \to 2} (3x^2 + 2x - 1) = 15$$ This example demonstrates that limits of polynomials can be evaluated by direct substitution due to their continuous nature.