Problem 23 - Calculus of a Single Variable

Question

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=5-x, g(x)=x^{3} $$ $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 1} f(x)} & {\text { (b) } \lim _{x \rightarrow 4} g(x)} & {\text { (c) } \lim _{x \rightarrow 1} g(f(x))}\end{array} $$

Step-by-Step Solution

Answer

The limits are: \(\lim _{x \rightarrow 1} f(x) = 4\), \(\lim _{x \rightarrow 4} g(x) = 64\), and \(\lim _{x \rightarrow 1} g(f(x)) = 64\)

Step 1: Evaluating \(\lim _{x \rightarrow 1} f(x)\)

In this step, the value of \(f(x)\) as \(x\) approaches 1 will be evaluated. Substituting \(x = 1\) in the function \(f(x) = 5 - x\), gives us \(f(1) = 5-1 = 4\). Thus, \(\lim _{x \rightarrow 1} f(x) = 4\).

Step 2: Evaluating \(\lim _{x \rightarrow 4} g(x)\)

In this step, the value of \(g(x)\) as \(x\) approaches 4 will be evaluated. Substituting \(x = 4\) in the function \(g(x) = x^{3}\), gives us \(g(4) = 4^{3} = 64\). Thus, \(\lim _{x \rightarrow 4} g(x) = 64\).

Step 3: Evaluating \(\lim _{x \rightarrow 1} g(f(x))\)

To find the limit of the composition of the functions, \(f(x)\) is plugged into \(g(x)\). This means that wherever \(x\) is in \(g(x)\), it's replaced with \(f(x)\). This results in \(g(f(x)) = (5-x)^{3}\). Substituting \(x = 1\) gives \((5 - 1)^3 = 4^3 = 64\). Thus, \(\lim _{x \rightarrow 1} g(f(x)) = 64\).

Key Concepts

Limit of a Function

The "limit of a function" refers to the value that a function approaches as the input approaches a certain point. It is a fundamental concept in calculus, often used to describe the behavior of functions near specific values. For example, when evaluating the limit \( \lim_{x \rightarrow a} f(x) \), you are finding what value \( f(x) \) gets closer to, as \( x \), the input, gets closer to \( a \). This process is crucial when dealing with continuous functions, where slight changes in \( x \) produce small changes in \( f(x) \).

To calculate limits, you can often substitute \( x \) with the desired value directly, assuming the function is continuous around that point. In our example, \( \lim_{x \rightarrow 1} f(x) \), we evaluated the function \( 5-x \) by simply substituting \( x = 1 \), resulting in the value 4. This demonstrates how straightforward finding limits can be when using well-behaved functions like linear or polynomial functions, providing the input is within the domain of the function.

Composition of Functions

The "composition of functions" occurs when one function is applied to the result of another function. With composition, you can build more complex functions from simple ones, and this can be denoted as \( g(f(x)) \). This means that for every \( x \), you first calculate \( f(x) \) and then use that result as the input for \( g(x) \).

In evaluating limits involving compositions like \( \lim_{x \rightarrow 1} g(f(x)) \), it's important to understand how each component function behaves independently as well as in sequence. In "Step 3," the result of \( f(x) = 5-x \) was plugged into \( g(x) = x^3 \). This transformed the problem into evaluating \( (5-1)^3 \), which simplified to \( 4^3 = 64 \). Hence, the limit of the composition \( g(f(x)) \) as \( x \rightarrow 1 \) is 64.

This showcases how knowing the behavior of individual functions aids in understanding their composition. It's essential to work carefully through compositions, especially for more complicated functions, as errors in initial function evaluations can lead to incorrect results.

Polynomial Functions

"Polynomial functions" are among the simplest and most foundational types of functions in calculus. They involve terms made up of coefficients and variables raised to positive integer exponents, such as \( a_nx^n + a_{n-1}x^{n-1} + \,...\, + a_0 \).

Polynomials are continuous and differentiable everywhere, making them predictable and manageable when finding limits. In our exercise, \( g(x) = x^3 \) is a cubic polynomial, which means it is continuous across all real numbers. Therefore, calculating \( \lim_{x \rightarrow 4} g(x) \) by simply substituting \( x = 4 \) into \( x^3 \) directly gives \( 4^3 = 64 \).

Polynomials are favored in calculus for their ease of manipulation and behavior prediction through their structure. Understanding polynomial functions' characteristics allows one to apply limits confidently, knowing they won't encounter undefined or discontinuous behavior within polynomials themselves.