Problem 8 - Thomas Calculus

Question

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$h(x)=\frac{3-(2 / x)}{4+\left(\sqrt{2} / x^{2}\right)}$$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{4} \) as \( x \rightarrow \infty \) and \( x \rightarrow -\infty \).

Step 1: Rewrite the Function

Start by rewriting the given function to separate terms containing \( x \) in the numerator and denominator: \[ h(x) = \frac{3 - \frac{2}{x}}{4 + \frac{\sqrt{2}}{x^2}}. \] This will help in finding the limits as \( x \) approaches infinity and negative infinity.

Step 2: Factor Out Dominant Terms

Factor out the dominant terms from both the numerator and the denominator. For \( x \rightarrow \infty \), the dominant term arises from constants since terms involving \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach zero: \[ h(x) = \frac{3 - 0}{4 + 0}. \]

Step 3: Evaluate Limit as \( x \rightarrow \infty \)

Evaluate the limit based on the simplified expression: As \( x \rightarrow \infty \), the function becomes \[ h(x) = \frac{3}{4}. \] So, the limit is \( \frac{3}{4} \).

Step 4: Evaluate Limit as \( x \rightarrow -\infty \)

Similar to \( x \rightarrow \infty \), for \( x \rightarrow -\infty \), terms involving \( \frac{1}{x} \) and \( \frac{1}{x^2} \) still approach zero. Repeating the same steps gives: \[ h(x) = \frac{3}{4}. \] Thus, the limit is still \( \frac{3}{4} \).

Key Concepts

Calculus

Calculus is a fundamental branch of mathematics that explores how things change. It primarily involves two types of operations: differentiation and integration. Differentiation deals with rates of change, such as the slope of a curve, while integration compiles an 'infinite' number of small pieces to find whole, such as an area under a curve.
Limits are an essential concept in calculus, used for defining both these operations more precisely. Understanding limits allows us to explore what happens to a function as its input approaches a specific value.

• For differentiation, limits help us find derivatives: the instantaneous rate of change or slope.
• In integration, limits help to sum infinitesimally small areas to find total area, volume, etc.

In the context of the exercise, limits help determine the behavior of the given function as it approaches infinity and negative infinity. They show how the function gradually stabilizes toward a particular value.

Infinity

Infinity is a concept rather than a number. It describes something endless or boundless. In calculus, infinity is commonly used to describe inexhaustible processes or unbounded behaviors.
When analyzing functions, the concept of infinity helps describe what happens as function inputs grow very large or very small (negative infinity). It allows us to explore quantities that otherwise seem too vast to work with conventionally.

• As a function approaches infinity, we observe its end behavior rather than its exact value.
• The idea allows us to conceptualize limits as the function’s input grows without bound.

In the presented exercise, taking the limit as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \) involves understanding how the terms depending on \( \frac{1}{x} \) and \( \frac{1}{x^2} \) diminish, simplifying the calculation of the function's limit.

Dominant Term Factorization

Dominant term factorization is a useful technique in limit evaluation, especially when dealing with functions that include variables in the denominator.
This technique involves identifying and focusing on the term with the highest power of the variable in polynomials within a function. This dominant term determines the character of the function's end behavior.

• Any term with diminishing influence (like those with \( \frac{1}{x} \) or \( \frac{1}{x^2} \)) becomes negligible.
• The approach simplifies the function, approximating its behavior at extreme ends of the variable input.

The exercise applies this by reducing more minor terms to zero, concentrating on constants in the solution. This highlights how the given function simplifies as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \), smoothly leading to the limit value \( \frac{3}{4} \).