Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 1}\left(2 x^{3}-6 x+5\right) $$

Short answer

The limit of the function \(2x^3 - 6x + 5\) as \(x\) approaches to 1 is 1.

Step-by-step solution

Identify the function

The function for which the limit is required to be found is \(2x^3 - 6x + 5\). The limit is to be evaluated as \(x\) approaches 1.

Substitute the limit point into the function

As this function is a polynomial, and is thus continuous everywhere, the limit at a point \(x = a\) is simply the function's value at that point. So, we only need to substitute the limit point \(x = 1\) into the function: \[(2(1)^3 - 6(1) + 5)\].

Calculate the limit

After substituting, we simplified the function: \(2(1) - 6 + 5 = 2 - 6 + 5 = 1\). Thus, the limit is 1.

Key concepts

Limits and Continuity

One of the foundational concepts in calculus is the idea of a limit. It’s a way of figuring out what value a function approaches as the argument gets closer and closer to a certain point. This concept is vital when considering the behavior of functions and is a stepping stone to understanding more complex ideas like derivatives and integrals.

Continuity, on the other hand, deals with how smooth a function is. If a function is continuous at a point, it means there's no abrupt change or gap in its graph at that point. In other words, you could draw it without lifting your pencil from the paper. For polynomials, which are combinations of variables raised to powers and multiplied by coefficients, continuity isn't an issue; they're continuous everywhere. This makes evaluating limits for polynomials at any given point straightforward, usually just requiring a direct substitution, as in our exercise.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of terms in the form of \( ax^n \) where \( n \) is a non-negative integer, and \( a \) is a coefficient. They are foundational in the study of calculus due to their simplicity and predictability.

Because of their continuous nature, polynomials don't exhibit jumps, holes, or vertical asymptotes — problems that can complicate limit evaluation in other types of functions. This characteristic implies that finding a limit as \( x \) approaches any value simply involves substituting that value into the polynomial to obtain the function's output, which is exactly what we see in the presented exercise where the limit is found by substituting \( x = 1 \) into the given polynomial.

Limit Evaluation

The process of evaluating limits can vary greatly depending on the type of function we're working with. For polynomial functions, the evaluation process is generally the simplest form of limit calculation. Since the function is continuous, you can often just substitute the value you're approaching directly into the function, which gives you the limit.

This direct substitution method, however, is just one technique in a toolbox of methods for evaluating limits that include factoring, rationalizing, or applying special limit laws. It’s important to note that more complex functions, especially those that are not continuous at a point, might require these other techniques. Utilizing direct substitution where applicable, as in the exercise we have, streamlines the process of finding limits and is an essential strategy to grasp for anyone studying calculus.