Question

How are $\underset{x\to a^{+}}{lim}f(x)$ and $\underset{x\to a^{+}}{lim}f(x)$ calculated if $f$ is a polynomial function?

Short answer

Question: Given a polynomial function, find the left and right limits as x approaches a value of 'a'. Answer: Since polynomial functions are continuous, the left and right limits as x approaches 'a' are both equal to the value of the function at 'a', i.e., f(a). To find the limits, simply substitute the value of 'a' into the polynomial function. Thus, $\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{+}}{lim}f(x)=f(a)$.

Step-by-step solution

Determine if the function f(x) is a polynomial function

First, confirm that the given function f(x) is indeed a polynomial function. This would mean that it is of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$, where $a_i$ are constant coefficients for each term, and n is a non-negative integer.

Evaluate the function at the value of 'a'

Since polynomial functions are continuous, just substitute the value of 'a' into the polynomial function to find the limit as x approaches 'a', regardless of the direction. In other words, find the value of $f(a)=a_n{a}^n+a_{n-1}{a}^{n-1}+...+a_{1}a+a_0$.

Determine the limits as x approaches 'a'

The left limit $\underset{x\to a^{-}}{lim}f(x)$ and the right limit $\underset{x\to a^{+}}{lim}f(x)$ are both equal to the value of the polynomial function at 'a', i.e., $f(a)$. Thus, $\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{+}}{lim}f(x)=f(a)$.

Key concepts

Continuity of Polynomials

Understanding the continuity of polynomial functions is crucial when working with limits. Polynomials are some of the smoothest functions you'll encounter in calculus. They're made up of terms that have variable bases raised to non-negative integer exponents, combined with constant coefficients. But what makes these functions continuous?

A function is continuous at a point if the left-hand limit, the right-hand limit, and the value of the function at that point are all equal. For polynomials, this is always true at every point in their domain, which is all real numbers. This means you can sketch the graph of a polynomial without lifting your pencil from the paper. Thanks to this property, evaluating a limit for a polynomial function is straightforward — you simply plug in the point you're approaching into the function. So, for any given polynomial function and any point 'a', the function will not have any gaps, jumps, or abrupt changes in direction at 'a' or indeed at any other point.

Evaluating Limits

Evaluating limits, especially for polynomials, is a fundamental concept in calculus. When you're asked to find $\underset{x\to a}{lim}f(x)$, you're essentially being asked what value the function approaches as 'x' gets infinitely close to 'a'. For polynomial functions, evaluating the limit as 'x' approaches 'a' can be done by direct substitution.

This process is possible because polynomials are continuous. If you're given $\underset{x\to a^{+}}{lim}f(x)$ or $\underset{x\to a^{-}}{lim}f(x)$, these notations are asking for the limit from the right or the limit from the left, respectively. However, since polynomials don't behave differently from the left or right, both of these limits will yield the same result as simply $f(a)$. This means that unless we have a more exotic kind of function, we don't need to worry about approaching from a particular direction — a huge simplification!

Polynomial Functions

Polynomial functions are like the building blocks of the calculus world. They are expressed in the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$, where $a_0,a_1,...,a_n$ are constants and 'n' is a non-negative integer representing the degree of the polynomial. The degree tells us the highest power of 'x' in the polynomial, and it is a big clue to the function's behavior and shape.

For instance, if 'n' is 2, the polynomial represents a parabola; if it's 3, it's a cubic curve, and so on. These functions are smooth and continuous everywhere, making them predictable and easy to graph. Each term in the polynomial affects the function's shape and curvature, and this interplay results in the diverse variety of polynomial graphs you'll encounter. Because they are so well-behaved, polynomials are often used to approximate more complex functions around a particular point, a technique foundational in calculus known as Taylor Series.