Question

How is \(\lim _{x \rightarrow a} f(x)\) calculated if \(f\) is a polynomial function?

Short answer

Question: Calculate the limit of the polynomial function \(f(x) = 5x^3 - 3x^2 + 2x - 7\) as \(x\) approaches 2. Answer: To calculate the limit of the polynomial function \(f(x) = 5x^3 - 3x^2 + 2x - 7\) as \(x\) approaches 2, we can directly substitute the value of 'a' (2) into the function: \(f(2) = 5(2)^3 - 3(2)^2 + 2(2) - 7 = 5(8) - 3(4) + 4 - 7 = 40 - 12 + 4 - 7 = 25\) Therefore, \(\lim _{x \rightarrow 2} f(x) = 25\).

Step-by-step solution

Understand Polynomial Functions

A polynomial function is a function consisting of variables, coefficients, and non-negative integer exponents. The general form of a polynomial function is: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ... , a_1, a_0\) are constants and \(n\) is a non-negative integer.

Understanding Limits

The limit of a function as x approaches a certain value (a) gives us the value that the function approaches when x gets arbitrarily close to (a), represented as \(\lim _{x \rightarrow a} f(x)\).

Limits of Polynomial Functions

For a polynomial function, the limit can be calculated directly by substituting the value of 'a' into the function: \(\lim _{x \rightarrow a} f(x) = f(a)\). This is because polynomial functions are continuous on their entire domain, which means that the value of the function at a point is equal to the limit of the function at that point.

Example Calculation

Let's say we have a polynomial function \(f(x) = 3x^2 - 2x + 4\), and we want to calculate \(\lim _{x \rightarrow 1} f(x)\). We can directly substitute the value of 'a' (1) into the function: \(f(1) = 3(1)^2 - 2(1) + 4 = 3 - 2 + 4 = 5\) Therefore, \(\lim _{x \rightarrow 1} f(x) = 5\).

Key concepts

Polynomial Functions

Polynomial functions are some of the most fundamental types of functions in mathematics. They are expressions built from variables, coefficients, and powers that are non-negative integers. You might often encounter them in forms like linear equations, quadratic equations, or even more complex formations. For a polynomial in its general form:

• It looks like this: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants.
• The leading constant \(a_n\) is called the leading coefficient, and it is crucial in determining the behavior of the polynomial, especially as \(x\) becomes very large or very small.
• The highest exponent, \(n\), is known as the degree of the polynomial and typically determines the general shape of the graph.

Polynomial functions are simple to work with because they are defined entirely by a finite number of coefficients, and operations like addition, subtraction, and multiplication involving them are straightforward.

Continuity in Calculus

The concept of continuity is central to calculus and especially relevant when discussing limits. A function is said to be continuous at a point if the value of the function at that point matches the value that the function approaches as we get infinitely close to that point. In simpler terms, there should be no holes, jumps, or breaks in the graph of the function. For polynomial functions:

• They are continuous everywhere on their domain, which is all real numbers. This means there are no interruptions or undefined points for typical polynomial functions; they smoothly curve across their entire graph.
• Because of their continuity, calculating limits for polynomials is as simple as substituting the given point directly into the function.
• Understanding continuity helps solidify the intuitive grasp on why certain functions behave predictably, which in turn makes solving calculus problems more straightforward.

In more complex functions, continuity can become a bit trickier, but with polynomials, it is very manageable and consistent, making them great introductory examples to study.

Limit Calculation Methods

Calculating the limit involves finding the value a function approaches as the input approaches a certain value. In polynomial functions, this process is particularly elegant. Here's how this works:

• The direct substitution method is used for polynomial functions because they are continuous, meaning \(\lim _{x \rightarrow a} f(x) = f(a)\).
• This simply involves inserting the value of \(a\), the point you are approaching, into the polynomial itself to find the limit.
• Let's take an example: for \(f(x) = 3x^2 - 2x + 4\), if you want to compute \(\lim _{x \rightarrow 1} f(x)\), you replace \(x\) with 1: \(f(1) = 3(1)^2 - 2(1) + 4 = 5\).

This method underscores the value of polynomial continuity and simplifies calculated outcomes. This contrasts more complex functions where limits might involve indeterminate forms or require techniques like L'Hôpital's Rule or trigonometric identities.