Problem 12 - Thomas Calculus

Question

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow 2}\left(-x^{2}+5 x-2\right)$$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Identify the Function

The given function for which we need to find the limit is \(-x^2 + 5x - 2\). Our goal is to find the limit as \(x\) approaches 2.

Step 2: Substitute the Limit Value

Substitute \(x = 2\) directly into the function. This step involves replacing \(x\) with 2 in the expression \(-x^2 + 5x - 2\).

Step 3: Simplify the Expression

Evaluate the expression by simplifying it:1. Substitute \(x = 2\): \(-x^2 + 5x - 2 = -(2)^2 + 5(2) - 2\).2. Calculate each term: \(-(2)^2 = -4\), \(5(2) = 10\).3. Combine the values: \(-4 + 10 - 2\).

Step 4: Calculate the Result

Combine the terms from the previous step:\(-4 + 10 - 2 = 4\). So, the limit as \(x\) approaches 2 is 4.

Key Concepts

Polynomial Function

Polynomial functions are mathematical expressions made up of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They are classified based on the highest power of the variable present in the function. For example, the function \[ f(x) = -x^2 + 5x - 2 \] seen in our exercise is a quadratic polynomial. This means the highest power of the variable \(x\) is 2.
Polynomial functions can take various forms, such as linear (power of 1), quadratic (power of 2), cubic (power of 3), and so on. They are continuous and smooth graphs, making them easy to work with in calculus since we can perform various operations like differentiation and integration.
Polynomials are crucial in calculus because they can approximate more complex curves, and limits of polynomial functions usually exist for any real number.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus used to analyze the behavior of a function as it approaches a certain point. Understanding limits lays the groundwork for computing derivatives and integrals. When we say we are evaluating a limit, we are determining what value a function approaches as the input approaches a certain number. In this exercise, we’re interested in how the polynomial \(-x^2 + 5x - 2\) behaves as \(x\) approaches 2.
There are several techniques for evaluating limits, such as

• direct substitution,
• factoring,
• rationalizing, and
• using L'Hopital’s Rule.

For polynomial functions, direct substitution is often the simplest method since these functions are continuous everywhere. If the substitution doesn't lead to an undefined form, the limit can be easily calculated.

Substitution Method

The substitution method is one of the simplest ways to evaluate limits, especially for polynomial functions. In this technique, we directly substitute the value that \(x\) is approaching into the function. Because polynomial functions are continuous, we can substitute this value without worrying about discontinuities or undefined outputs.
Let's walk through the substitution method for the exercise:

• Identify the point \(x\) is approaching, which is 2 in our case.
• Substitute \(x = 2\) into the polynomial \(-x^2 + 5x - 2\),resulting in: \[-(2)^2 + 5(2) - 2.\]
• Simplify to \[-4 + 10 - 2,\]leading to a final result of 4.

Thus, for this polynomial function, the limit as \(x\) approaches 2 is 4. This shows how the substitution method provides a quick way to find limits for such expressions.