Question

If $$ G(x)=c_{0}+c_{1} x^{2}+c_{2} x^{2}+\cdots+c_{n} x^{n} $$ then \(\lim _{x \rightarrow a} G(x)=G(a)=c_{0}+c_{1} a+c_{2} a^{2}+\cdots+c_{n} a^{n}\).

Short answer

Answer: The limit of the polynomial function G(x) as x approaches the value 'a' is \(G(a) = c_0 + c_1a + c_2a^2 + \cdots + c_na^n\).

Step-by-step solution

Identify the form of the polynomial function G(x)

Given the polynomial function: $$ G(x) = c_0 + c_1x + c_2x^2 + \cdots + c_nx^n $$

Evaluate G(a)

We are asked to find \(\lim_{x \rightarrow a} G(x)\). To do this, we will substitute x = a into the polynomial function G(x) and calculate G(a): $$ G(a) = c_0 + c_1a + c_2a^2 + \cdots + c_na^n $$

Determine the limit of G(x) as x approaches a

Since G(x) is a polynomial function, the limit of G(x) as x approaches a is given by evaluating G(a). Therefore, we have: $$ \lim_{x \rightarrow a} G(x) = G(a) = c_0 + c_1a + c_2a^2 + \cdots + c_na^n $$ This is the final expression for the limit of the polynomial function G(x) as x approaches the value 'a'.

Key concepts

Polynomial Function

A polynomial function is a type of function characterized by a sum of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. In more technical terms, it's an expression of the form ewline ewline ewline ewline ewline For example, if we consider the polynomial function given in our exercise, ewline ewline ewline ewline As we can see, each term has a different power of x, starting from x raised to the 0th power (which is 1) and potentially increasing to higher powers. The coefficients ( are constants that can be any real numbers.

The Importance of Polynomial Functions

Polynomials are foundational for various fields of mathematics and science. They help describe a wide range of phenomena like trajectories in physics or growth models in biology. Their simple structure makes computing with them easily understandable and manageable for students, all the while still being broadly applicable.

Calculating Limits

When we talk about calculating limits, we're discussing how to determine the value that a function approaches as the input (or variable) gets closer to a certain point. Limits are crucial in calculus and mathematical analysis, forming the basis of notions such as continuity, derivatives, and integrals.

Understanding Limits

For polynomial functions, calculating limits is often straightforward. If we're dealing with a polynomial function and we want to find the limit as x approaches a particular value, we can usually just plug this value into the function. This straightforward method is possible because polynomial functions are continuous everywhere—they don’t have breaks, jumps, or 'holes'.

Substitution Method

The substitution method is one of the simplest and most direct approaches to finding limits. It's based on the concept that if a function is continuous at a certain point, you can calculate the limit by substituting the point directly into the function.

Applying the Substitution Method

To apply the substitution method, simply replace x with the value it is approaching in the function's equation. In the case of the polynomial given in the exercise we would find the limit as x approaches 'a' by calculating This demonstrates the straightforward application of the substitution method for continuous functions like polynomials.

Properties of Limits

The properties of limits refer to the rules that allow us to calculate limits more effectively and understand their behavior. These properties are backed by theoretical principles which ensure that limits are consistent and calculable under certain conditions.

Key Properties of Limits

• Limit of a Sum: The limit of a sum is equal to the sum of the limits.
• Limit of a Product: The limit of a product is equal to the product of the limits.
• Limit of a Constant: The limit of a constant is the constant itself.
• Continuous Functions: If a function is continuous at a certain point, then the limit as we approach that point is just the function value at that point.

By using these properties, one can solve more complex limit problems and grasp the behavior of functions as they approach different values. For polynomial functions, which are continuous everywhere, the limit as x approaches any number is just the value of the polynomial at that number, reaffirming their simplicity and continuity.