Question

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=2 x^{2}-3 x+1, g(x)=\sqrt[3]{x+6} $$ $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 4} f(x)} & {\text { (b) } \lim _{x \rightarrow 1} g(x)} & {\text { (c) } \lim _{x \rightarrow 4} g(f(x))}\end{array} $$

Short answer

The solution is: (a) 23, (b) \(\sqrt[3]{7}\), (c) \(\sqrt[3]{29}\)

Step-by-step solution

Solve for (a) \(\lim _{x \rightarrow 4} f(x)\)

Substitute \(x = 4\) into the function \(f(x)\) to find the limit. We get \(f(4) = 2(4)^2 - 3(4) + 1 = 23\). Therefore, \(\lim _{x \rightarrow 4} f(x) = 23\).

Solve for (b) \(\lim _{x \rightarrow 1} g(x)\)

Substitute \(x = 1\) into the function \(g(x)\) to find the limit. We get \(g(1) = \sqrt[3]{1 + 6} = \sqrt[3]{7}\). Therefore, \(\lim _{x \rightarrow 1} g(x) = \sqrt[3]{7}\).

Solve for (c) \(\lim _{x \rightarrow 4} g(f(x))\)

First, substitute \(x = 4\) into the function \(f(x)\) to find the new value that will be used in function \(g(x)\). From the first step, we know that \(f(4) = 23\). Therefore, the function \(g(f(4))\) is equal to \(g(23) = \sqrt[3]{23 + 6} = \sqrt[3]{29}\). Therefore, \(\lim _{x \rightarrow 4} g(f(x)) = \sqrt[3]{29}\).

Key concepts

Continuous Functions

A continuous function, in simple terms, is a function that you can draw on a graph without lifting your pencil off the paper. These functions do not have any breaks, jumps, or discontinuities. Continuity is an essential idea in calculus, enabling us to analyze behaviors of functions reliably.
To check for continuity at a particular point, we use three criteria:

• The function must be defined at the point.
• The limit of the function as it approaches the point must exist.
• The value of the function at the point must equal the limit as it approaches the point.

In our exercise, when we evaluate limits by substitution, it implies the functions are continuous at the given values. Thus, substitutes work smoothly without causing undefined behaviors or discontinuities.

Polynomial Functions

Polynomial functions, like the one given by \(f(x) = 2x^2 - 3x + 1\), are expressions consisting of variables and coefficients. These are often called algebraic expressions of degree \(n\) where no term has a degree greater than \(n\).
They are inherently continuous over the entire set of real numbers, making them very predictable and stable when calculating limits. This is because polynomial functions do not have any abrupt jumps or vertical asymptotes.
To find the limit of a polynomial function as \(x\) approaches some value, we simply substitute that value into the function, as we did in our solution finding the limit \(\lim_{x \to 4} f(x)\).

Composition of Functions

The composition of functions involves combining two functions to create a new one. If we have two functions, \(f\) and \(g\), the composition is denoted as \(g(f(x))\). This means you first apply function \(f\) to \(x\), then take that result and apply function \(g\) to it.
This technique is particularly useful for solving complex problems as it allows us to layer processes. In our exercise, we first found \(f(x)\) for \(x = 4\), and then used that output as the input for \(g(x)\). The composed function \(g(f(x))\) allows us to manage step-by-step transformations effectively.
Understanding this composition is vital for derivatives and integrals, which are foundational concepts in calculus.

Limits of Composite Functions

Limits of composite functions deal with finding the limit of a function like \(g(f(x))\) as \(x\) approaches a certain value. The strategy here is to first find the limit of the inner function, and then apply it to the outer function.
In our example, we found \(\lim_{x \to 4} f(x) = 23\) and then used this to determine \(g(23)\).
When both functions \(g\) and \(f\) are continuous, as is often the case here, the process simplifies as continuity ensures there are no unexpected results. Hence, we successfully calculated \(\lim_{x \to 4} g(f(x)) = \sqrt[3]{29}\).
This concept is important for understanding more intricate mathematical scenarios where multiple functions interact.