Problem 93 - Calculus of a Single Variable

Question

Using the Definition of Limits at Infinity Consider $$\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}+3}}$$ (a) Use the definition of limits at infinity to find values of \(M\) (b) Use the definition of limits at infinity to find values of \(M\) that correspond to \(\varepsilon=0.1 .\)

Step-by-Step Solution

Answer

The limit as x approaches infinity of \(\frac{3x}{\sqrt{x^{2}+3}}\) is 3. The specific values of \(M\) depend on the value of \(ε\)

Step 1: Analyzing the function

We first simplify the function by dividing the numerator and the denominator of the function by \(x^{2}\). The simplified function then becomes \(\frac{3}{\sqrt{1+(3/x^2)}}\) as \(x\) approaches infinity. Here, the \(3/x^{2}\) term goes to 0 as \(x\) approaches infinity, the function becomes \(\frac{3}{\sqrt{1+0}} = 3\)

Step 2: Calculating M for general case

The definition of limits at infinity states that for every positive number \(ε\), there is a positive number \(M\) such that \(|f(x) - L| < ε\) for all \(x > M\), where \(L\) is the limit of the function as \(x\) approaches infinity. Here, \(L = 3\) and \(f(x) = \frac{3}{\sqrt{1+(3/x^2)}}\). So, \(|f(x) - L| = |\frac{3}{\sqrt{1+(3/x^2)}} - 3| \). Therefore, to find \(M\), we solve the inequality \(|\frac{3}{\sqrt{1+(3/x^2)}} - 3| < ε\) for \(x\). Then, \(M\) is any positive number greater than the solution of the inequality.

Step 3: Calculating M for ε=0.1

We need to solve the inequality \(|\frac{3}{\sqrt{1+(3/x^2)}} - 3| < 0.1\) for \(x\). The solution of the inequality provides the value of \(x\) where \(|\frac{3}{\sqrt{1+(3/x^2)}} - 3|\) is less than \(0.1\). Then, \(M\) is any positive number greater than the solution of the inequality.

Key Concepts

Definition of Limits

The concept of limits, especially as they approach infinity, is a cornerstone of calculus. When we say we want to find the limit of a function as it approaches infinity, we are looking at what happens to the function's values as the inputs become very large.

Imagine an airplane flying towards the horizon; as it goes further away, it nears the line of the horizon, which can be thought of as a limit. However, the plane never quite touches the horizon, just as function values approach the limit without excessively exceeding it.

In mathematical terms, the limit of a function \( f(x) \) as \( x \) approaches infinity is often written as \( \lim_{x \to \infty} f(x) \). This expression means that we are examining what the function value gets closer to as the input \( x \) grows larger and larger.

ε-M Definition

The \( \varepsilon \)-\( M \) definition is a formal way to express limits at infinity. It describes how as \( x \) becomes extremely large, the value of the function \( f(x) \) will stay very close to a limiting value \( L \).

You can think of \( \varepsilon \) (epsilon) as a small positive number that signifies how close we want the function to be to the limit \( L \). \( M \) is a threshold beyond which all function values are within \( \varepsilon \) of \( L \).

Here’s how it works in practice:

• For every \( \varepsilon > 0 \), there exists a number \( M > 0 \).
• If \( x > M \), then the inequality \(|f(x) - L| < \varepsilon\) holds true.

This condition ensures that beyond some large value of \( x \), the function does not deviate from \( L \) by more than an amount \( \varepsilon \).

Function Simplification

Simplifying functions before calculating limits is a crucial step, as it often unveils the behaviors hidden in their complexity. In the problem at hand, the original function \( \frac{3x}{\sqrt{x^2 + 3}} \) becomes far more manageable through simplification.

The goal is to transform the function into one where the influence of \( x \) becoming infinitely large does not obscure the approach towards the limit. By dividing both numerator and denominator by \( x \), we focus on the dominant terms at infinity:

• The numerator simplifies to \( \frac{3}{1} \) since any term involving \( \frac{1}{x} \) will become negligible as \( x \to \infty \).
• The denominator, after the simplification \( \sqrt{x^2(1 + \frac{3}{x^2})} \), becomes \( \sqrt{1 + \frac{3}{x^2}} \), further simplifying the expression.

As a result, this process reveals that the function approaches a clear constant, making it easier to determine the limit.

Calculating Limits

Once our function is simplified, calculating limits involves determining what value the function approaches. For the given problem, once the function \( f(x) = \frac{3x}{\sqrt{x^2 + 3}} \) is simplified to \( \frac{3}{\sqrt{1 + \frac{3}{x^2}}} \), we consider \( x \to \infty \).

As \( x \) increases, the term \( \frac{3}{x^2} \) becomes negligible, approximating the function as \( \frac{3}{\sqrt{1 + 0}} = 3 \). Thus, the limit \( L \) is \( 3 \).

To find real values for \( M \) that satisfy \( \varepsilon = 0.1 \), we solve the inequality \(|f(x) - 3| < 0.1\). After calculating, we determine the minimum \( x \) value beyond which the deviation from 3 is less than 0.1. Practically, \( M \) is any number larger than this critical \( x \) value, ensuring the function stays close to its limit of 3.