Problem 22 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}+1}} $$

Step-by-Step Solution

Answer

The limit of the given function as \(x\) approaches negative infinity is \(-1\).

Step 1: Simplify the expression

Begin by making \(x\) negative. Replace \(x\) with \(-x\). We get: \(\frac{-x}{\sqrt{(-x)^2+1}}\).

Step 2: Factoring out -x from the expression

Factoring out \(-x\) from the numerator and the denominator term by term we get: \(-x\left(\frac{1}{\sqrt{1+\frac{1}{x^{2}}}}\right)\). This allows us to separate the variables \(x\) into two parts.

Step 3: Find the limit as \(x\) approaches negative infinity

We know that for square roots, as the denominator inside the square root becomes increasingly larger, the entire denominator approaches zero. Since as \(x\) goes to negative infinity, the term \(\frac{1}{x^{2}}\) goes to zero. Thus the limit becomes: \(-1*\frac{1}{\sqrt{1+0}}=-1\).

Key Concepts

Calculus

Calculus is an essential branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It is used to study the changes between values that are related by a function. In the broader scope, calculus helps us to understand the dynamics of systems and processes in various scientific, economic, and engineering domains.

With calculus, we can calculate how quickly something is moving at a precise instant (instantaneous rate of change), the total accumulation of something over an interval (area under a curve), and other similar concepts. Understanding the principles of calculus is crucial for tackling complex problems that involve precise calculations of change and accumulation.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input approaches a particular value. Limits help us understand a function's value as the input gets infinitely close to a number, which may not be attained by the function itself. This becomes especially important when dealing with discontinuities or points at which the function is not defined.

For instance, in the exercise where we need to find the limit \( \lim _{x \rightarrow -\infty} \frac{x}{\sqrt{x^{2}+1}} \), we are concerned with how the function \( \frac{x}{\sqrt{x^{2}+1}} \) behaves as \( x \) becomes increasingly negative without bound. By understanding limits, we gather that certain terms become negligible in the presence of very large numbers (or small, when approaching zero), and the function's behavior can be determined by the dominant terms, leading to a conclusion about the function's asymptotic behavior.

Asymptotic Behavior

Asymptotic behavior in calculus refers to the tendencies of functions as the input either grows very large (positive or negative infinity) or becomes very small. It is about understanding how a function behaves at the extreme ends of the domain or what value it approaches. This characteristic is often associated with the 'end-behavior' of a function.

In the provided exercise, as we take the limit of \( \frac{x}{\sqrt{x^{2}+1}} \) as \( x \) approaches negative infinity, we observe that the square root term in the denominator becomes increasingly large, which makes the whole fraction's absolute value shrink towards zero. However, since the fraction involves a negative numerator (negative \( x \) over positive square root), this yields a limit of -1. Understanding asymptotic behavior is key in predicting how a function behaves in the long run, which is vital in many fields like physics, engineering, and economics.