Problem 48 - Precalculus

Question

Evaluate the limits using the limit properties. $$\lim _{x \rightarrow 2}\left(x^{4}-2 x+5\right)$$

Step-by-Step Solution

Answer

The limit is 17.

Step 1: Identify the limit expression

The given limit expression is \(\lim _{x \rightarrow 2}\left(x^{4}-2 x+5\right)\). Our task is to evaluate this limit as \(x\) approaches 2.

Step 2: Apply the limit property for polynomials

Since the expression inside the limit is a polynomial, we can directly substitute the value of \(x\) to find the limit. The limit properties for polynomials permit the direct substitution method as these functions are continuous.

Step 3: Substitute \(x = 2\) into the polynomial

Substitute \(x = 2\) into the polynomial function: \(x^4 - 2x + 5\). This gives \(2^4 - 2(2) + 5\).

Step 4: Calculate the power and products

Compute the terms in the expression by first calculating the power and products: \(2^4 = 16\), \(2 \times 2 = 4\).

Step 5: Sum up the results

Add the calculated values together: \(16 - 4 + 5\), which simplifies to \(17\).

Step 6: Conclusion: State the evaluated limit

The limit of the expression as \(x\) approaches 2 is \(17\).

Key Concepts

Limit Properties

Limits are a fundamental concept in calculus and are used to describe the behavior of functions as they approach a certain point. Limit properties allow us to simplify and evaluate limits, especially if the function behaves nicely, as is often the case with polynomials.

There are several important properties of limits that can be useful:

• **Addition/Subtraction**: The limit of a sum/difference is the sum/difference of the limits, i.e., \(\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)\).
• **Multiplication**: The limit of a product is the product of the limits, meaning \(\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\).
• **Division**: The limit of a quotient is the quotient of the limits (given the limit of the denominator isn't zero):\(\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}\).
• **Constant**: The limit of a constant is just the constant itself, \(\lim_{x \to c} a = a\).

Using these properties simplifies complex expressions, especially when working with continuous functions. They also establish that the limit operation distributes over many algebraic operations, like addition and multiplication.

Polynomial Functions

Polynomial functions are composed of variables raised to whole number powers and combined using addition, subtraction, and multiplication. A general expression for a polynomial is \a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0,\ where \(a_n, a_{n-1}, \, ... \, , a_0\) are constants.

These functions have certain distinct characteristics:

• **Continuity**: Polynomial functions are continuous over all real numbers, meaning there are no breaks, holes, or jumps in their graphs.
• **Differentiable**: Polynomials are differentiable everywhere, making them smooth and easy to work with regarding calculus operations, like finding maxima/minima.
• **End Behavior**: The highest degree term's behavior often dominates the polynomial's behavior as \(x\) moves toward positive or negative infinity.

Due to their continuity and smooth behavior, evaluating the limits of polynomial functions is straightforward. You can simply substitute the value into the polynomial to find the limit as \(x\) approaches a specific value.

Direct Substitution Method

The direct substitution method is a straightforward way to evaluate limits, especially handy when dealing with continuous functions like polynomials. This method involves substituting the value that \(x\) approaches directly into the function.

For example, when working with \(\lim_{x \to 2} (x^4 - 2x + 5)\), the direct substitution method allows you to replace \(x\) with \(2\) right away because polynomial functions do not have any discontinuities.

Follow these easy steps:

• **Identify the Point of Approach**: Determine the value \(x\) approaches; in our example, it is 2.
• **Substitute**: Replace \(x\) with the point of approach in the expression, giving \(2^4 - 2 \times 2 + 5\).
• **Simplify**: Calculate each term using basic arithmetic (like powers, products, and sums) to find the limit value.

Using these simple steps for polynomial functions will help evaluate the limit quickly without complications. This approach proves valuable for most limits involving continuous functions.