91Ó°ÊÓ

The concept of the limit of a function is essential in calculus and helps us understand the behavior of functions as the input approaches a specific value. In simpler terms, when we talk about the limit of a function, we are asking what happens to the value of the function as we get closer and closer to a certain point on the x-axis.
To determine the limit of a function, say for the function \( f(x) = x^2 + 1 \) as \( x \) approaches \( -1 \), we calculate how \( f(x) \) behaves as \( x \) gets very close to \( -1 \). We substitute values closer and closer to \( -1 \) and observe the trend of the function's output. If our calculated limit approaches a real number, we say that the limit exists.
For the given exercise, the limit was found to be 2 when \( x \) approaches \( -1 \). This tells us that as \( x \) gets infinitesimally close to \( -1 \), the value of \( f(x) \) nears 2. It is crucial for the limit to exist to consider whether a function is continuous at a point.

Function evaluation is the process through which we find the output of a function for a specific input value. This is typically done by substituting the input value into the function's formula.
For the function \( f(x) = x^2 + 1 \), evaluating the function at \( x = -1 \) means substituting \( -1 \) into the function, resulting in:

• \( f(-1) = (-1)^2 + 1 \)
• \( f(-1) = 1 + 1 \)
• \( f(-1) = 2 \)

This calculation shows us that the function's output at \( x = -1 \) is 2. Moreover, verifying that the function is defined at \( x = -1 \) is crucial in determining continuity. Function evaluation is a basic yet important task that allows us to explore many mathematical properties, including checking for continuity, as seen in this exercise.

Polynomial functions are sums of terms consisting of a variable raised to an exponent and multiplied by a coefficient. These functions are of the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \). For our function, \( f(x) = x^2 + 1 \), it is a simple polynomial of degree 2, often referred to as a quadratic function.
Polynomial functions, like \( f(x) = x^2 + 1 \), have many interesting properties:

• They are smooth and continuous over all real numbers.
• They do not have any gaps, jumps, or infinite oscillations.
• They have a predictable behavior and can easily be evaluated for derivatives and integrals.

The function \( f(x) = x^2 + 1 \) is continuous at any value of \( x \), including \( x = -1 \), as verified in the original exercise. This takes us to the broader concept of continuity for polynomial functions, which is guaranteed by their inherent construction and properties.