Question

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0}\left(4 x^{2}-5 x-8\right)$$

Short answer

\(-8\)

Step-by-step solution

Identify the Limit

Understand that the exercise is asking for the limit of the function as x approaches 0. The function given is a polynomial.

Plug in the Value of x

Since the function is a polynomial and thus continuous everywhere, we can straightforwardly substitute x = 0 into the function to find the limit.

Calculate the Limit

Replace x with 0 in the polynomial and simplify: \(4(0)^2 - 5(0) - 8 = -8\). This gives us the limit.

Key concepts

Calculating Limits

Understanding how to calculate limits in calculus is a foundational skill that allows us to determine what value a function approaches as the input gets infinitely close to a certain number. To calculate a limit, we often substitute the approach value directly into the function, provided the function is continuous at that point. If the function is not continuous, we might need to apply other methods, such as factoring, conjugates, or L'Hôpital's Rule.

In the given exercise, the limit is calculated as the input variable, in this case, 'x', approaches zero. Since it's a polynomial function which is inherently continuous, we can substitute zero directly into the equation to find that the limit is \( -8 \). Such direct evaluation is usually the first step to try when calculating limits, as it provides a quick and often correct answer.

Limits Involving Zero

When a limit involves the variable approaching zero, it can signify reaching a function's value at a particular point or evaluating the behavior near an undefined or non-existent point. It's critical to look out for forms that could indicate discontinuities or indeterminate forms such as \( \frac{0}{0} \) or \( 0 \cdot \infty \).

In polynomial functions, such as the exercise's example, a limit involving zero is straightforward because polynomials do not have discontinuities or indeterminate forms. You can simply plug in zero in place of the variable to find the limit value, which leads to calculation without encountering complexity or ambiguity.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients structured in terms of non-negative integer power. A significant property of these functions is continuity; they are continuous for all real numbers. This simply means that you can draw the graph of a polynomial without lifting your pencil from the paper.

Properties of Polynomial Functions:

• They have smooth, continuous curves.
• There are no sharp corners or cusps.
• They can be represented by the standard form \( ax^n + bx^{n-1} + \dots + k \) where \( n \) is a non-negative integer and \( a, b, \dots, k \) are constants.

These properties make polynomial functions approachable when calculating limits, as seen in the example where the function \( 4x^2 - 5x - 8 \) maintains continuity at zero, ensuring that calculating the limit is a matter of simple substitution.

Continuous Functions

Continuous functions are those functions that have no interruptions, holes, jumps, or vertical asymptotes in their graphs. The concept of continuity is crucial in calculus as it ensures that limits can be calculated by direct substitution. If a function \( f(x) \) is continuous at a point \( c \) then \( \lim_{x \rightarrow c} f(x) = f(c) \).

Most common algebraic functions, including polynomials, rational, root, trigonometric, exponential, and logarithmic functions, can be continuous over their domains. For instance, the polynomial in the exercise is a continuous function, which means as \( x \) approaches zero, we can find the limit by evaluating the function at that point. Consequently, continuity simplifies the process of finding limits and serves as an essential tool for analyzing function behaviors near specific points.