Problem 5 - Calculus Single Variable

Question

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow+\infty} \frac{x+\sqrt{x}}{x^{2}-\sqrt{x}} $$

Step-by-Step Solution

Answer

The limit exists and equals 0.

Step 1: Identify the expression form

The given limit is \( \lim _{x \rightarrow+\infty} \frac{x+\sqrt{x}}{x^{2}-\sqrt{x}} \). This fraction is composed of a numerator and a denominator, both of which are polynomial-like expressions involving \(x\).

Step 2: Determine dominant terms

For \( x \to +\infty \), compare the terms in both the numerator and the denominator. In the numerator \( x + \sqrt{x} \), the dominant term is \( x \). In the denominator \( x^2 - \sqrt{x} \), the dominant term is \( x^2 \).

Step 3: Simplify the expression

Divide both the numerator and the denominator by the dominant terms to simplify. The dominant term for division is \( x^2 \), simplifying the expression to: \(\lim _{x \to +\infty} \frac{\frac{x}{x^2} + \frac{\sqrt{x}}{x^2}}{1 - \frac{\sqrt{x}}{x^2}}\).

Step 4: Simplify further with limits

Calculate the limits of the simplified terms. As \( x \to +\infty \), \( \frac{x}{x^2} \to \frac{1}{x} \to 0 \), \( \frac{\sqrt{x}}{x^2} \to \frac{1}{x^{3/2}} \to 0 \), and \( \frac{\sqrt{x}}{x^2} \to \frac{1}{x^{3/2}} \to 0 \). The expression simplifies to \( \lim _{x \to +\infty} \frac{0 + 0}{1 - 0} \).

Step 5: Conclude the limit value

The simplified expression \( \frac{0}{1} \) resolves to 0, meaning the limit exists and its value is 0. Thus, \( \lim _{x \rightarrow+\infty} \frac{x+\sqrt{x}}{x^{2}-\sqrt{x}} = 0 \).

Key Concepts

Dominant Term

When dealing with limits involving polynomial expressions, understanding the concept of the dominant term is crucial. The dominant term is the part of an algebraic expression that grows the fastest as the variable, usually denoted as \( x \), approaches infinity or negative infinity. It dictates the behavior of the entire expression under these conditions. In our exercise, the expression is \( \lim_{x \rightarrow +\infty} \frac{x + \sqrt{x}}{x^2 - \sqrt{x}} \). To find the dominant term, look at the terms with the highest power of \( x \) in both the numerator and the denominator.

• For the numerator \( x + \sqrt{x} \), the dominant term is \( x \) since \( x \) grows faster than \( \sqrt{x} \) as \( x \) increases.
• For the denominator \( x^2 - \sqrt{x} \), the dominant term is \( x^2 \), because \( x^2 \) grows significantly faster than \( \sqrt{x} \).

Recognizing these dominant terms helps in simplifying the process of finding limits, as they primarily influence the behavior of the function for very large or very small values of \( x \).

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the context of limits, especially as \( x \) approaches infinity, polynomial expressions play a pivotal role in determining the behavior of a function.In our problem, we deal with the polynomial-like expressions in the form \( x + \sqrt{x} \) and \( x^2 - \sqrt{x} \). These are not pure polynomials since they include \( \sqrt{x} \), but they behave similarly when considering limits:

• The expression in the numerator is primarily influenced by the term \( x \).
• The expression in the denominator is dominated by the term \( x^2 \).

To solve limits with such expressions, identifying the polynomial components and factoring out the dominant terms helps simplify the expression, focusing on the most influential parts. This simplification makes it easier to calculate the limit, as the less dominant terms become insignificant as \( x \) approaches infinity.

Infinity in Limits

Understanding infinity in limits is essential for mastering calculus. Infinity describes a quantity without bound or end. When considering limits as \( x \) approaches infinity, the goal is to determine how a function behaves as \( x \) gets indefinitely large. In our exercise, \( \lim_{x \rightarrow +\infty} \frac{x + \sqrt{x}}{x^2 - \sqrt{x}} \)describes the ratio of two expressions as \( x \) becomes very large.The approach to solving such problems involves:

• Identifying dominant terms that have the greatest influence on the function.
• Simplifying the expression using these dominant terms to understand what happens as \( x \to +\infty \).

For example, in our given limit, as \( x \) grows, less significant terms (like \( \sqrt{x} \) relative to \( x \) and \( x^2 \)) diminish to insignificance. This method leaves us with simpler forms, hence in our case the limit simplifies to \( \frac{0}{1} = 0 \). Understanding the behavior of limits as \( x \) approaches infinity is crucial for analyzing the long-term behavior of functions in calculus.