Problem 84 - Calculus: Early Transcendentals

Question

A logarithm limit a. Draw a graph to verify that \(-|x| \leq x^{2} \ln x^{2} \leq|x|\), for \(-1 \leq x \leq 1,\) where \(x \neq 0\). b. Use the inequality in part (a) to evaluate \(\lim x^{2} \ln x^{2}\).

Step-by-Step Solution

Answer

Question: Verify the inequality \(-|x| \leq x^{2} \ln x^{2} \leq |x|\) for \(-1 \leq x \leq 1,\) where \(x \neq 0\), and then use the Squeeze Theorem to find the limit \(\lim_{x\to 0} x^{2} \ln x^{2}\). Answer: The given inequality is verified, as the function \(y=x^{2}\ln x^{2}\) is bounded between the functions \(y=-|x|\) and \(y=|x|\) in the interval \(-1\leq x\leq1\), with \(x\neq0\). Applying the Squeeze Theorem and knowing the limits of the bounding functions equal 0 as \(x\to0\), we find that the limit \(\lim_{x\to 0} x^{2} \ln x^{2} = 0\).

Step 1: Draw the graphs of the functions

Plot the three functions: \(y=-|x|\), \(y=x^{2} \ln x^{2}\), and \(y=|x|\) on the same graph for the interval \(-1\leq x\leq1\), while keeping in mind that \(x\neq0\). Notice that in the given interval, the middle function is bounded between the other two functions.

Step 2: Use the Squeeze Theorem

The Squeeze Theorem states that if we have functions \(f(x)\), \(g(x)\), and \(h(x)\), and if \(f(x)\leq g(x) \leq h(x)\) when \(a < x < b\), and if the limits of the bounding functions are the same, i.e., \(\lim_{x \to c}f(x)=\lim_{x \to c}h(x)=L\), then \(\lim_{x \to c}g(x)=L\). In our case, we have the inequality \(-|x| \leq x^{2} \ln x^{2} \leq |x|\) for \(-1\leq x\leq1\). We want to find the limit \(\lim_{x\to0}x^{2}\ln x^{2}\).

Step 3: Evaluate the limits of the bounding functions

Let's evaluate the limits of the bounding functions \(f(x)=-|x|\) and \(h(x)=|x|\): 1. \(\lim_{x\to0}-|x|=0\) 2. \(\lim_{x\to0}|x|=0\) Both limits equal 0.

Step 4: Apply the Squeeze Theorem

Since we have the inequality \(-|x| \leq x^{2} \ln x^{2} \leq |x|\) and the limits of the bounding functions as \(x \to 0\) both equal 0, we can use the Squeeze Theorem: $$\lim_{x \to 0} x^{2} \ln x^{2} = 0.$$ Thus, the limit of the function \(y=x^{2} \ln x^{2}\) as \(x\to0\) is 0.

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a fundamental concept in Calculus that helps us affirm a limit when a function is "squeezed" between two others. Let's think about the basic principle: suppose we have three functions, specifically let's call them \(f(x)\), \(g(x)\), and \(h(x)\). If these functions satisfy the condition \(f(x) \leq g(x) \leq h(x)\) at points very close to \(x = c\) (aside from possibly at \(c\) itself), and if

• \(\lim_{x \to c} f(x) = L\)
• \(\lim_{x \to c} h(x) = L\)

then the limit of \(g(x)\) will also be \(L\). This theorem is especially useful when the function \(g(x)\) is complicated or tricky to evaluate directly. In our exercise, \(x^2 \ln x^2\) is squeezed between \(-|x|\) and \(|x|\). To employ the Squeeze Theorem, we examined the limits of these bounding functions as \(x\) approached zero, and since both limits were zero, so was the limit of \(x^2 \ln x^2\). This is an elegant example of the Squeeze Theorem in action!

Limits

A limit in calculus describes the value that a function approaches as the input approaches a certain point. Understanding limits is crucial because it forms the base of understanding continuity, derivatives, and integrals. In this context, finding a limit involves considering how a function behaves as the variable \(x\) gets very close to a specific value, often 0, or towards infinity.

In our problem, we are interested in evaluating the limit \(\lim_{x \to 0} x^2 \ln x^2\). This usually requires understanding the behavior of the factor \(x^2\) as it goes to zero and how it interacts with the logarithmic part \(\ln x^2\), which diverges. However, by using the Squeeze Theorem, we look at other functions that "trap" \(x^2 \ln x^2\) in a range that helps to evaluate its actual limit more effortlessly.

When both bounding functions surrounding a tricky function share a limit value, it provides an insightful and easier method to evaluate otherwise difficult limits. This is one reason why studying the behavior of functions as they stretch towards infinity or shrink towards zero is such a core exercise in calculus.

Logarithms

Logarithms are the inverse operation to exponentiation and are incredibly useful in the world of calculus. The logarithmic function, especially the natural logarithm \(\ln(x)\), translates multiplication into addition and division into subtraction. This property can simplify handling products and powers immensely.

In this exercise, the term \(x^2 \ln x^2\) involves a logarithm. As \(x\) approaches zero, the logarithm tends to negative infinity which reflects one of the fascinating characteristics of logarithms; they grow slower and slower, diverging slower than polynomial or exponential functions visible in other contexts.

Understanding the logarithmic function helps in visualizing and sketching graphs that show how processes like these differ profoundly from polynomial ones, especially around limits. This skill is vital when exploring bounds and constraints, as it underscores the essential dynamics ruled by exponential growth and decay.

Inequalities

Inequalities are a way to express the relative size or order of two values. Instead of claiming equality, inequalities suggest that one quantity could be larger, smaller, or possibly equal. In calculus, inequalities are essential for understanding concepts such as range and bounding of functions.

In our scenario, the inequality \(-|x| \leq x^2 \ln x^2 \leq |x|\) serves as a structure that helps us use the Squeeze Theorem. This inequality basically captures that the function \(x^2 \ln x^2\) is constrained within two other functions. This simple yet powerful inequality insight sets the scene for determining limits without calculating difficult expressions directly.

Inequalities like these often indicate boundaries of behavior and validate whether certain conditions are met, such as showing that a function is within a stable "sandwich" to find its behavior as its input reaches a limit. Overall, using inequalities in calculus fosters and simplifies mathematical estimations and can efficiently direct an analysis towards the concluding insights.