Problem 25 - Calculus Early Transcendentals

Question

Evaluate the following limits. \(\lim _{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+4 x+5\right)\)

Step-by-Step Solution

Answer

Question: Evaluate the limit of the function \(f(x) = 2x^3 - 3x^2 + 4x + 5\) as \(x\) approaches 1. Answer: The limit of the function as \(x\) approaches 1 is 8.

Step 1: Identify the polynomial function

We are given the polynomial function \(f(x) = 2x^3 - 3x^2 + 4x + 5\) and asked to evaluate the limit as \(x \rightarrow 1\).

Step 2: Calculate the limit by plugging in x = 1

Since polynomial functions are continuous everywhere, we can directly evaluate the limit by plugging in the value x = 1 into the function: \(f(1) = 2(1)^3 - 3(1)^2 + 4(1) + 5\)

Step 3: Simplify the expression

Now, we simplify the expression to find the limit: \(f(1) = 2(1) - 3(1) + 4 + 5\) \(f(1) = 2 - 3 + 4 + 5\) \(f(1) = 8\)

Step 4: Write the answer

The evaluation of the limit is: \(\lim _{x \rightarrow 1}(2 x^{3}-3 x^{2}+4 x+5) = 8\)

Key Concepts

Limits of Polynomial Functions

Understanding the concept of limits is crucial for students as it serves as the fundamental building block in calculus. Specifically, when it comes to polynomial functions, such as the given function \( f(x) = 2x^3 - 3x^2 + 4x + 5 \), evaluating limits becomes an intuitive process due to the friendly nature of polynomials.

Polynomial functions are made up of terms that include variables raised to whole number exponents, like \( x^3 \) or \( x^2 \) in our example, and constants. One important property of polynomial functions is that they are continuous for all real numbers. This means that there are no breaks, holes, or jumps in the graph of a polynomial function. Continuous functions allow the limit as \( x \) approaches any real number to be the same as the function's value at that number.

Therefore, to evaluate the limit of a polynomial function as \( x \) approaches any given value, one can typically use direct substitution, a simple and practical method that often delivers the answer swiftly and without complexity.

Continuity of Polynomials

One might wonder why polynomial functions are guaranteed to be continuous everywhere. This comes from the nature of their construction. Since polynomials are comprised of one or more monomials added together, and each monomial is continuous on its own, the sum of these monomials (the polynomial) inherits this continuity.

Continuity in mathematical terms means that small changes in the input, or \( x \), result in small and predictable changes to the output, or \( f(x) \). It implies there are no abrupt changes in the value of the function. For all polynomial functions—whether they're as simple as \( x^2 \) or as complex as \( 10x^{15} - 4x^7 + x \)—you can graph them without lifting your pencil off the paper.

Why is Continuity Important in Calculus?

Continuity plays a pivotal role in calculus because it ensures the reliability of various operations and analyses, such as differentiation and the evaluation of limits themselves. In the case of polynomial functions, their continuity implies the lack of need for complicated limit evaluation techniques and thus simplifies our approach to solving calculus problems.

Direct Substitution in Limits

Direct substitution is a method often utilized to evaluate limits, especially with polynomial functions. It is exactly what it sounds like: directly substituting the value that \( x \) approaches into the function. In our example, the limit as \( x \) approaches 1 of the function \( f(x) = 2x^3 - 3x^2 + 4x + 5 \) is found by simply replacing every \( x \) in the polynomial with 1.

The effectiveness of this technique lies in its simplicity and the properties of the function in question. Because there are no discontinuities in polynomials, as explained earlier, the limit as \( x \) approaches any particular value is exactly the same as the polynomial's value when \( x \) is equal to that particular value.

When Direct Substitution Doesn't Apply

However, it's essential to recognize that direct substitution is not the go-to method for all functions. When dealing with non-polynomial functions, particularly those with discontinuities or undefined points at the value of interest, you would need to deploy other techniques to evaluate the limit, such as factoring, rationalizing, or applying L'Hôpital's rule.