Question

What does it means to say that limits of polynomials may be evaluated by direct substitution?

Short answer

Question: Evaluate the limit of the polynomial function f(x) = x^3 - 4x^2 + 5x - 2 as x approaches 1 using direct substitution. Answer: The limit of the polynomial function as x approaches 1 is 0.

Step-by-step solution

Identify the polynomial function

We have the polynomial function f(x) = x^3 - 4x^2 + 5x - 2. Step 2: Write down the limit equation

Write down the limit equation

We want to find the limit as x approaches 1, written as lim_(x->1) f(x). Step 3: Direct substitution

Direct substitution

Since we are dealing with a polynomial function, we can find the limit by direct substitution. In this case, we will substitute x = 1 into the polynomial function: f(1) = (1)^3 - 4(1)^2 + 5(1) - 2. Step 4: Simplify the expression

Simplify the expression

Simplify f(1) = (1) - 4(1) + 5(1) - 2 = 1 - 4 + 5 - 2. Step 5: Calculate the limit

Calculate the limit

The limit is equal to the simplified expression: lim_(x->1) f(x) = 1 - 4 + 5 - 2 = 0. Therefore, it means that the limit of the given polynomial function as x approaches 1 is 0.

Key concepts

Direct Substitution

Direct substitution is a straightforward technique used in calculus to evaluate the limit of a polynomial function. The process involves taking the value that the variable approaches and plugging it directly into the polynomial. This is possible because polynomials are continuous functions; they have no sudden breaks or holes in their graphs for any real value of x.

If a limit exists at a certain point, direct substitution will give you the exact value of that limit. For example, suppose we are tasked with finding \( \lim_{x \to 1} f(x) \) for the function \( f(x) = x^3 - 4x^2 + 5x - 2 \). By substituting \( x = 1 \) into the function, we can directly evaluate the limit. This simplifies to \( f(1) = 1 - 4 + 5 - 2 \), which further simplifies to 0, giving us the limit at that point.

Polynomial Functions

Polynomial functions are algebraic expressions that involve a sum of powers of the variable x, where all exponents are whole numbers, and the coefficients are real numbers. These functions are widely appreciated for their simplicity and the ease with which they can be worked with algebraically and graphically.

Basic examples of polynomials include \( x^2 - 4x + 7 \) and \( 3x^3 + x - 5 \). Polynomials can be graphed easily and exhibit smooth and continuous curves without any sharp corners or discontinuities. This is one reason why the technique of direct substitution works seamlessly with them when evaluating limits.

Calculus

Calculus is a branch of mathematics that focuses on change and motion. It is divided mainly into two areas: differential calculus and integral calculus. Differential calculus concerns the concept of a derivative, which measures how a function changes as its inputs change. On the other hand, integral calculus focuses on the concept of an integral, which represents the accumulation of quantities and can be thought of as the area under a curve.

Limits are one of the fundamental concepts in calculus and serve as the foundation for defining derivatives and integrals. They help us understand the behavior of functions as they approach specific points or infinity.

Limits Evaluation

In calculus, evaluating limits is a critical skill that allows us to determine the behavior of functions as they approach a particular value. Techniques for evaluating limits include direct substitution, factorial expansion, limits at infinity, and L'Hôpital's Rule, among others. While some functions allow for direct substitution, others that have indeterminate forms require more advanced strategies for limit evaluation.

Understanding when and how to apply these various techniques is crucial for correctly solving calculus problems. As we've seen with polynomial functions, the direct substitution method is often the most efficient way to find the limit of a function at a point where the function is continuous.