Problem 80 - Single Variable Calculus

Question

Let \(f(x)=\frac{x+2}{\sqrt[4]{x+18}-2}\) a. Plot the graph of \(f\), and use it to estimate the value of \(\lim _{x \rightarrow-2} f(x)\) b. Construct a table of values of \(f(x)\) accurate to three decimal places, and use it to estimate \(\lim _{x \rightarrow-2} f(x)\). c. Find the exact value of \(\lim _{x \rightarrow-2} f(x)\) analytically. Hint: Make the substitution \(x+18=t^{4}\), and observe that \(t \rightarrow 2\) as \(x \rightarrow-2\).

Step-by-Step Solution

Answer

The exact value of the limit \(\lim_{x \rightarrow -2} f(x)\) is 24, which can be found analytically using the substitution \(x + 18 = t^4\) and rewriting the function in terms of \(t\). This limit can also be estimated visually from the graph and by constructing a table of values for the function.

Step 1: Graph the Function

First, we will plot the graph of \(f(x) = \frac{x + 2}{\sqrt[4]{x + 18}-2}\). By plotting the graph and observing how the curve behaves as \(x\) approaches -2, we can get an estimate of the limit.

Step 2: Estimate the Limit Visually

By looking at the graph, observe the value of the function \(f(x)\) as \(x\) gets closer and closer to -2. Estimate the value of the limit \(\lim _{x \rightarrow-2} f(x)\) by determining to what value the function appears to be getting closer.

Step 3: Table of Values

Create a table of values for the function \(f(x)\) around the point \(x = -2\) with increasing precision, and round them to three decimal places. Use these values to estimate the limit \(\lim _{x \rightarrow -2} f(x)\). Example: | x | f(x) | |-------|---------| | -2.1 | | | -2.01 | | | -2.001| | Fill in the table with values of f(x) and observe how the values converge to a particular number.

Step 4: Substitution and Exact Value

To find the exact value of the limit, follow the hint and make the substitution \(x + 18 = t^4\). Rewrite the function in terms of \(t\), and observe that as \(x \rightarrow -2\), \(t \rightarrow 2\). Then, find the limit analytically. Since \(x + 18 = t^4\), we have \(x = t^4 - 18\) and \(x + 2 = t^4 - 16\). Substituting into the function, we get: \(f(t) = \frac{t^4 - 16}{\sqrt[4]{t^4} - 2}\) Using the fact that \(\sqrt[4]{t^4} = t\), we get: \(f(t) = \frac{t^4 - 16}{t - 2}\) Now, as \(x \rightarrow -2\), \(t \rightarrow 2\), so we want to find \(\lim_{t \rightarrow 2} f(t)\). Use polynomial division or factoring to rewrite the function: \(f(t) = \frac{(t - 2)(t^3 + 2t^2 + 4t + 8)}{t - 2}\) Now cancel the term \((t-2)\) and find the limit: \(\lim_{t \rightarrow 2} f(t) = \lim_{t \rightarrow 2} (t^3 + 2t^2 + 4t + 8)\) Finally, substitute \(t = 2\) to get the exact value of the limit: \(\lim_{t \rightarrow 2} f(t) = (2^3 + 2\cdot2^2 + 4\cdot2 + 8) = \boxed{24}\) Therefore, the exact value of the limit \(\lim_{x \rightarrow -2} f(x) = 24\).

Key Concepts

Function Graphing

Graphing a function provides an important visual tool to understand how the function behaves. In this case, the exercise asks us to explore the behavior of the function \( f(x) = \frac{x+2}{\sqrt[4]{x+18}-2} \) as \( x \) approaches -2. Here’s how graphing assists us:

• When you plot the graph, look closely at the part of the curve near \( x = -2 \). The graph gives a visual cue of where the function values are heading.
• The graph approach helps in estimating the limit without performing detailed calculations.

The visual nature of graphing can make abstract limit concepts more concrete and easier to digest. You can use graphing software or calculators to aid this process if drawing by hand is not convenient.

Table of Values

Utilizing a table of values is an effective technique to numerically estimate limits. It involves selecting \( x \) values closer and closer to your point of interest, \( x = -2 \), and calculating the corresponding \( f(x) \) values.

• The idea is to systematically approach \( x = -2 \) from both the left and right sides, narrowing in on the target value.
• For instance, you may choose values like -2.1, -2.01, and -2.001, and observe how \( f(x) \) behaves as it approaches the limit from those positions.

This method allows students to see how the function values evolve, helping reinforce their understanding of limits in a structured manner. Moreover, rounding values to three decimal places in this context enhances precision and comparison.

Substitution Method

In solving limits analytically, substitutions can simplify complex expressions. The exercise suggests substituting \( x + 18 = t^4 \), transforming our function into a new variable \( t \).

• Substitution makes the function easier to manipulate by changing variables that better handle incoming behavior as the limit is approached.
• Here, \( t \to 2 \) as \( x \to -2 \) guides the substitution steps.

Rewriting \( f(x) \) in terms of \( t \) activates new problem-solving strategies, like factoring or simplifying by rewriting components. Substitution is powerful for revealing underlying structures that facilitate solving complex calculus problems.

Polynomial Division

Polynomial division is used to simplify expressions, particularly in limit calculations, by eliminating indeterminate forms. In this problem, after substitution yields \( f(t) = \frac{t^4 - 16}{t - 2} \), we employ polynomial division.

• The goal is to rewrite the function such that the indeterminate form \((t-2)\) in the denominator cancels out with a similar factor in the numerator.
• Once canceled, it leaves a polynomial that is easier to evaluate.

This process eliminates complexities and provides the avenue to finally substitute \( t = 2 \) directly, achieving a precise computation of the limit as \( t \to 2 \), which results in solving the limit accurately.