Question

Consider \(\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(M\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).

Short answer

The limit of the function as x approaches infinity is 3. For the given values of \(\varepsilon\), the inequality from the definition of limit is satisfied for all values of \(M\), hence any \(M\) can be chosen.

Step-by-step solution

Simplify the function

Simplify the function by dividing numerator and denominator of the function by \(x\), which leaves us with: \[\lim _{x \rightarrow \infty} \frac{3}{\sqrt{1+\frac{3}{x^{2}}}}\] As \(x\rightarrow\infty\), \(\frac{3}{x^{2}}\) tends to 0, hence the function simplifies to \[\lim _{x \rightarrow \infty} \frac{3}{\sqrt{1+0}} = 3\]

Definition of limit at infinity

The definition of limit at infinity states that for any given \(\varepsilon\), there exists an \(M\) such that if \(x > M\) then \(|f(x) - L| < \varepsilon\). Here \(L=3\).

Find values of \(M\) for given \(\varepsilon\)

a) For \(\varepsilon=0.5\), from the definition of limit, we have |3 - L| < 0.5. Because \(L=3\), the inequality becomes |0| < 0.5 which is true for all \(M\), hence any \(M\) can be chosen. \nb) For \(\varepsilon=0.1\), similarly, |3 - L| < 0.1. The inequality is true for all \(M\), hence any \(M\) can be chosen.

Key concepts

Calculus

Calculus is the branch of mathematics that studies changes and rates of change. It can be broadly divided into two main areas: differential calculus and integral calculus. Differential calculus involves finding the rate at which quantities change, known as derivatives. Integral calculus, on the other hand, deals with the accumulation of quantities, such as areas under curves or the total growth calculated from a rate of change, and these are called integrals.

In the world of calculus, understanding the behavior of functions as they approach certain points or infinity is crucial. This is where the concept of limits becomes a foundational aspect of the discipline. Limits allow for the rigorous analysis of functions at points where they might not be clearly defined, such as at points of discontinuity or as they extend towards infinity.

Limits are also fundamental in proving various rules and theorems in calculus, such as the derivative and integral rules. A strong grasp of limits and their properties enables students to dive deeper into mathematical analysis and solve complex problems involving instantaneous rates and total accumulations.

Limit Definition

When exploring the behavior of functions, the definition of a limit is the tool that encapsulates how a function behaves as its argument gets infinitely close to some point — or, in our case, approaches infinity. In formal terms, the limit of a function as 'x' approaches a certain value 'a' represents the value that 'f(x)' approaches as 'x' draws nearer to 'a'.

When discussing limits at infinity, we are concerned with what value 'f(x)' tends towards as 'x' grows larger and larger without bound. For instance, in the given exercise, the limit is defined as the number 'L' that the function \frac{3}{\brm{\brm{1 + \frac{3}{x^{2}}}}} tends to as 'x' goes to infinity. The meticulous part is choosing how close to 'L' we wish to be, which is our '\(\epsilon\)', and determining the boundary 'M' after which our function will always be within '\(\epsilon\)' of 'L'.

This definition can be applied to various kinds of functions to understand their end-behavior, making it an essential concept in calculus for analyzing the growth, decay, or stabilization of functions as they extend towards the great unknown—infinity.

Asymptotic Behavior

The term 'asymptotic behavior' refers to the way functions behave as they approach a line or curve ever so closely without necessarily touching it, as their input becomes incredibly large or small. In many cases, this line or curve is called an 'asymptote'. The concept of limits at infinity is commonly used to identify these asymptotic behaviors.

As an example, consider the function in the exercise where 'x' approaches infinity. By defining the limit, we can conclude that the function comes increasingly close to the value 3, implying that the horizontal line 'y = 3' acts as a horizontal asymptote to the function. Asymptotic behavior is a way to describe this long-term behavior succinctly.

Understanding asymptotes is crucial for graphing functions and predicting their long-term trends. However, not all functions exhibit asymptotic behavior as they do not all settle towards a single value at infinity. Recognizing the presence or absence of this behavior is a valuable skill when analyzing functions and their graphs in calculus.