Question

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

Short answer

To find the limit of a polynomial function as x approaches a given value a, apply the properties of the limit such as the limit of a constant function c as x approaches any value a is c, the limit of x as x approaches a value a is a, and the limit of a product of functions is equal to the product of their limits. Then, compute the limit of individual terms, sum them up to get the overall limit of the polynomial function, and finally simplify the expression to find the limit of the given polynomial function as x approaches a value a.

Step-by-step solution

Definition of a Polynomial Function

A polynomial function is a function of the form $$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $$ where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants and \(n\) is a non-negative integer.

Properties of the Limit of a Polynomial Function

Limit of a polynomial function has the following properties: 1. The limit of a constant function c as x approaches any value a is c. 2. The limit of x as x approaches a value a is a. 3. The limit of a polynomial function as x approaches a, behaves as a sum of its individual terms. 4. The limit of a product of functions is equal to the product of their limits. Using these properties, we can find the limit of a polynomial function as x approaches a certain value a.

Identify the polynomial function and the value of a

Write down the given polynomial function \(f(x)\) and the value of x that it approaches (a).

Apply limit properties

For the given polynomial function \(f(x)\), apply the limit properties mentioned above, to get the limit of each individual term as x approaches a.

Compute the limit of individual terms

For each term, compute the limit as x approaches a using the limit properties.

Sum the limits of individual terms

Once you have calculated the limits of each individual term, sum them up to get the overall limit of the polynomial function: $$ \lim _{x \rightarrow a} f(x) = \lim _{x \rightarrow a} (a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0) $$

Simplify and conclude

Finally, simplify the overall limit expression to find the limit of the given polynomial function as x approaches a value a.