Question

How are \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\) calculated if \(f\) is a polynomial function?

Short answer

Question: Explain how to calculate the left-hand limit and right-hand limit of a polynomial function as x approaches a specific point. Answer: To calculate the left-hand limit (\(\lim_{x\rightarrow a^{-}} f(x)\)) and right-hand limit (\(\lim_{x\rightarrow a^{+}} f(x)\)) of a polynomial function, substitute the point \(a\) directly into the function, since polynomial functions are continuous everywhere. Therefore, both the left-hand limit and right-hand limit at any point \(a\) are equal to the function's value at that point. In other words, \(\lim_{x\rightarrow a^{-}} f(x) = f(a)\) and \(\lim_{x\rightarrow a^{+}} f(x) = f(a)\).

Step-by-step solution

Understanding Polynomial Functions

A polynomial function is a function of the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where \(a_n, a_{n-1}, \cdots, a_1, a_0\) are constants and \(n\) is a non-negative integer. Polynomial functions are continuous everywhere, meaning that they have no breaks, jumps, or holes in their graphs. This property will help us when determining the left-hand limit and right-hand limit of a polynomial function.

Understanding Left-hand and Right-hand Limits

Left-hand limit (\(\lim_{x\rightarrow a^{-}} f(x)\)) refers to the limit of the function \(f(x)\) as \(x\) approaches point \(a\) from the left side. Similarly, right-hand limit (\(\lim_{x\rightarrow a^{+}} f(x)\)) refers to the limit of the function as \(x\) approaches point \(a\) from the right side.

Calculating the Left-hand and Right-hand Limits of a Polynomial Function

Since polynomial functions are continuous everywhere, their left-hand limit and right-hand limit at any point \(a\) are equal to the function's value at that point. Therefore, we can directly substitute the point \(a\) into the function to find the left-hand limit and right-hand limit. For a polynomial function: \(\lim_{x\rightarrow a^{-}} f(x) = f(a)\) \(\lim_{x\rightarrow a^{+}} f(x) = f(a)\) So, the left-hand limit and the right-hand limit of a polynomial function are equal to each other, and calculating them involves simply plugging the point \(a\) into the polynomial function.

Key concepts

Limits

In mathematics, limits help us understand the behavior of functions as they approach a particular point or value. They give us insight into how a function behaves close to a specific input. When dealing with polynomial functions, limits become relatively straightforward. Due to the continuous nature of polynomials, you can find the limit by simply evaluating the function at the given point. If you're asked to find \( \lim_{x\rightarrow a} f(x) \), you would substitute \( x = a \) into the polynomial to find its value. This simplicity stems from the fact that polynomial functions are smooth and unbroken at all points in their domain.

Continuity

Continuity in a function means there are no interruptions in its graph. Polynomial functions are a great example of continuous functions because they have no breaks, holes, or jumps. This property ensures that the value of a polynomial function at any given point is the same from both directions. Hence, when you're finding limits, the path you choose to approach the point doesn't affect the result. The continuity of polynomial functions makes them easy to work with in calculus, especially when computing limits. When you know a function is continuous, like polynomials, you don't need to worry about tricky discontinuities.

Left-hand Limit

The left-hand limit refers to the value that a function approaches as the input comes from the left towards a specific point. Mathematically, it's expressed as \( \lim_{x\rightarrow a^{-}} f(x) \).For polynomial functions, finding the left-hand limit is straightforward. Since they are continuous everywhere they are defined, the left-hand limit is the same as the function's value at that point. To compute it, simply substitute the point into the polynomial and evaluate. Even though it entails approaching from the left, continuity ensures no difference from the calculated function value.

Right-hand Limit

The right-hand limit is the value a function approaches as the input comes from the right towards a specific point. It's represented as \( \lim_{x\rightarrow a^{+}} f(x) \).Because polynomial functions lack breaks or jumps, their right-hand limit also coincides with the actual value of the function at the point. This fact simplifies calculations greatly because the right-hand limit, like the left-hand limit, just requires evaluating the polynomial at the given point.In polynomial functions, both left-hand and right-hand limits being equal to the function's value serves as a reassuring check of continuity, a vital trait in calculus.