Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Find the limit. $$ \lim _{x \rightarrow 1}\left(1-x^{2}\right) $$

Step-by-Step Solution

Answer

The limit of the function \(1 - x^2\) as \(x\) approaches 1 is 0.

Step 1: Understanding the limit function

You are given a function \(1 - x^2\) and are asked to find its limit as \(x\) approaches 1.\n The limit of a function at a certain point is the value that the function approaches as the variable \(x\) approaches that point.

Step 2: Substituting the Limit Value

For the given function, substitute \(x\) with 1, which is the value \(x\) is approaching. The function becomes \(1 - (1)^2\).

Step 3: Calculate the Limit

Now continue the calculation as follows: \(1 - (1)^2 = 1 - 1 = 0\)

Key Concepts

Algebraic functions

Algebraic functions are mathematical expressions formed by a combination of numbers, variables, and operations like addition, subtraction, multiplication, division, and taking roots. These functions can be simply expressed or involve more complex interactions between elements. For example, functions like polynomials are a common type of algebraic functions.

In the given exercise, the function is expressed as \(1 - x^2\). This is a simple algebraic function that involves subtraction and squaring operations. Understanding the behavior and composition of algebraic functions is essential as they frequently appear in calculus and are fundamental to evaluating limits.

The function \(1 - x^2\) can be identified as a polynomial, which is a common form of an algebraic function. These are useful as they are continuous and differentiable, making the process of finding limits more straightforward.

Evaluating limits

Evaluating limits involves determining the value that a function approaches as the variable within it reaches a specified point. The concept of a limit is foundational in calculus as it helps in analyzing the behavior of functions close to a certain input, even if the function doesn’t actually reach that point.

For our exercise, we want to find \( \lim_{x \to 1} (1 - x^2) \). To evaluate this limit, you substitute the value towards which \(x\) is approaching into the function. In most cases for algebraic functions, if substituting directly results in a valid expression, the process is straightforward.

Here, by plugging in \(x = 1\), the expression becomes \(1 - (1)^2\), simplifying directly to zero. This clean calculation results from the continuous nature of the algebraic function, which often allows direct substitution to find limits.

Limit properties

Limit properties are rules that help simplify the process of finding limits, particularly when dealing with complex expressions. They work best with algebraic operations and functions, providing a way to break down and analyze limits using known mathematical principles.

Some fundamental properties include:

• The limit of a sum is the sum of the limits: \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
• The limit of a difference is the difference of the limits: \( \lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \)
• The limit of a product is the product of the limits, and the same for quotients if the divisor's limit isn't zero.
• The limit of a constant times a function is the constant times the limit of the function.

In our case, the limit of \(1 - x^2\) as \(x\) approaches 1, highlights the property of the limit of a difference. By applying this principle, we see the function simplifies directly to zero without further breakdown, thus showing how these properties lead to an effortless evaluation for simple polynomial functions.