Problem 28 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The limit of the function \(-x^{2}+x-2\) as \(x\) approaches 2 is \(-4\).

Step 1: Identify the Limiting Value

The first step is to identify the value that \(x\) is approaching. In this case, it is given that \(x\) is approaching 2. This value will be substituted into the given function.

Step 2: Substitution

Substitute 2 into the function \(-x^{2}+x-2\). So, you have \(-2^{2}+2-2\).

Step 3: Evaluation

Next, evaluate the expression. You will then find that it equals \(-4+2-2\) which simplifies to \(-4\).

Key Concepts

Polynomial Functions

Polynomial functions are a special kind of mathematical expression. They consist of variables raised to whole-number exponents, alongside numerical coefficients. Specifically, polynomial functions are of the form:

• \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where each \( a_i \) is a constant and \( n \) is a non-negative integer.
This structure makes them versatile and able to model many real-world scenarios.
In the exercise, the given function is a polynomial because it contains terms like \(-x^2\) and \(+x\), arranged with whole-number powers and coefficients. Polynomial functions such as this one are smooth and continuous, which means they have no sudden jumps or breaks. This smoothness greatly aids in finding limits, as they behave predictably within the space they occupy.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus. It helps us understand the behavior of functions as the input values approach a particular number.
When we talk about the 'limit' of a function as \( x \) approaches a certain value, we are observing what value the function's output gets closer to, without actually reaching the point. This is expressed as \( \lim_{x \to a} f(x) \).
For polynomial functions, evaluating limits is straightforward because they are continuous.

• You simply need to analyze the function's behavior at specific points or through interval mapping.
• Understanding limits can visualize how functions behave at boundaries or when nearing singular points.

This can tell you whether the function settles on a particular value or shifts into undefined areas, although the latter case seldom applies to basic polynomial functions.

Substitution Method

The substitution method is a practical technique for evaluating limits of polynomial functions.
It's very efficient and involves replacing the variable in the function with the value that \( x \) is approaching.
In our exercise, where we find the limit of \( \lim _{x \to 2}(-x^2 + x - 2) \), the process demonstrates the substitution method in action:

• First, replace \( x \) with 2: this turns the expression into \(-2^2 + 2 - 2 \).
• Then, simply calculate the arithmetic: \(-4 + 2 - 2 \), which equals \(-4\).

The substitution method is particularly effective for polynomial functions because of their continuity and lack of undefined points. This makes it usually possible to directly substitute the limit point without encountering indeterminate forms.