Problem 10 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 2}\left(-x^{3}+1\right) $$

Step-by-Step Solution

Answer

-7

Step 1: Substitute \(x = 2\) into the equation

Substitute \(x = 2\) in the equation \(-x^{3}+1\), which becomes \(-2^{3}+1\).

Step 2: Calculate the power

Calculate \(2^{3}\), which becomes 8. Substitute into the equation and we get \(-8+1\).

Step 3: Calculate the final result

Calculate \(-8+1\), which becomes -7.

Key Concepts

Substitution Method for Limits

The substitution method is a straightforward way of finding limits of a function as the variable approaches a certain value. In this method, the main idea is to directly substitute the value into the function. This approach works when the function is defined at the point or approaches a finite value.

• First, identify the limit you need to find, as in \(\lim_{x \to 2} (-x^3 + 1)\).
• Check if substituting the value directly into the function leads to a valid outcome without indeterminate forms like \(\frac{0}{0}\).
• If it does, substitute the value directly and compute the result.

In our exercise, substituting \(x = 2\) into \(-x^3 + 1\) transforms the equation into \(-2^3 + 1\). This method is especially effective for simple polynomials and continuous functions.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole number powers and their coefficients. They are one of the simplest forms of functions to work with in calculus because they are continuous and differentiable everywhere.

• Polynomials include terms like \(ax^n\), where \(n\) is a non-negative integer and \(a\) is a constant coefficient.
• A simple example is \(f(x) = -x^3 + 1\), which is a third-degree polynomial.
• Polynomials have predictable behavior at limits, making them simple to evaluate using substitution.

In \(-x^3 + 1\), we see typical polynomial behavior. As the degree of the polynomial is 3, the function is continuous and the substitution method can be directly applied to find limits at any point.

Calculating Limits

Finding limits is a fundamental concept in calculus, representing the value that a function approaches as the variable gets infinitely close to a given point. Calculating limits can often be done using simple substitution for straightforward functions.

• The process involves evaluating the function at or very close to the point of interest.
• Look for any discontinuities or indeterminate forms, which may require different techniques like factoring or rationalization.
• Once substitution is possible, calculate the terms step-by-step as shown in the original exercise.

In our example, after substituting \(x = 2\) into \(-x^3 + 1\), we compute \(-2^3 + 1\) to find the limit as \(-7\). Calculating limits in this manner involves straightforward arithmetic, leading to clear and concise results.