Question

In Exercises \(101-104,\) use the definition of limits at infinity to prove the limit. $$ \lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0 $$

Short answer

Using the definition of the limit at infinity, it can be shown that, for any given \( M > 0 \), there exists a number \( N = \frac{4}{M^2} \), such that whenever \( x > N \), \( |\frac{2}{\sqrt{x}} - 0| < M \). This verifies that indeed \(\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0\).

Step-by-step solution

Understand the Definition of Limit at Infinity

A function \( f(x) \) has a limit \( L \) at infinity if, for any number \( M \) in real numbers, there exists a number \( N \) such that, for all \( x \) with \( x > N \), it implies that \( |f(x) - L| < M \). Here, the function is \( f(x) = \frac{2}{\sqrt{x}} \) and \( L = 0 \).

Apply the Definition

Let's proceed with the problem by using the logic from the definition: for all \( M > 0 \), find \( N \) satisfying the condition. Since the limit \( L = 0 \), the condition thus becomes \( |\frac{2}{\sqrt{x}} - 0| < M \) which simplifies to \( |\frac{2}{\sqrt{x}}| < M \).

Simplify the Inequality

The inequality \( |\frac{2}{\sqrt{x}}| < M \) translates to \( \frac{2}{\sqrt{x}} < M \), since \( \frac{2}{\sqrt{x}} \) is always positive for real \( x \). Squaring both sides to get rid of the square root gives \( \frac{4}{x} < M^2. \) Solving for \( x \), we get \( x > \frac{4}{M^2} = N.\)

Conclusion

So, for any \( M > 0 \), one can always find a number \( N = \frac{4}{M^2} \), such that whenever \( x > N \), \( |\frac{2}{\sqrt{x}} - 0| < M \). This verifies the definition of the limit at infinity. Hence \(\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0\).

Key concepts

Definition of Limit at Infinity

Understanding the definition of limit at infinity is crucial for grasping many concepts in calculus. Simply put, it entails finding the behavior of a function as the input value grows without bounds. In formal terms, we say that a function f(x) approaches a limit L as x approaches infinity, if for any positive number ε, no matter how small, there exists a corresponding large positive number N such that for all x > N, the value of |f(x) - L| is less than ε. This notion helps us understand how a function behaves at the extreme ends of its domain.

For example, in the exercise \( \lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0 \), we're essentially looking at how close the function f(x) = \frac{2}{\sqrt{x}} gets to zero as x becomes arbitrarily large. Breaking down the problem using this definition allows students to verify and understand the limit's validity rigorously.

Proving Limits

Proving limits is a systematic process that demonstrates the validity of a limit statement. To prove a limit, we need to use logical reasoning and mathematical manipulations strictly adhering to the definitions and properties of limits. The textbook exercise showcases this process in action. The student leverages the formal definition of a limit at infinity by selecting an ε and finding a corresponding N that meets the criteria for every value after it.

The importance of careful argumentation cannot be overstated as it strengthens the understanding and ensures the soundness of the calculus performed. Proving a limit as in the step-by-step solution demonstrates mastery of the concept and gears students up for more complex problems in calculus.

Infinite Limits of Functions

Infinite limits of functions describe situations where a function's value increases or decreases without any bound as the input approaches some value, or as the input itself grows or shrinks without bound. In contrast to limits that approach a finite number, an infinite limit indicates that a function can grow or decrease indefinitely. When the limit notation includes infinity as a value, either as the target limit or as part of the limit expression, it signals that we're dealing with an unbounded behavior.

While the textbook exercise involves a limit that approaches zero, other scenarios could involve a function f(x) where the limit goes to infinity as x approaches a certain value, or a negative infinity if the function decreases without bound. It's essential to be precise when handling infinity in these contexts to avoid misconceptions and ensure accuracy.

Solving Inequalities in Calculus

The step-by-step solution provided in the textbook exercise integrates another core concept of calculus: solving inequalities. When evaluating limits, particularly at infinity, inequalities can help us estimate the behavior of functions and establish the necessary bounds for proof. By manipulating inequalities, similar to equations, one can isolate the variable of interest and find suitable values that satisfy the given conditions.

In the textbook example, the inequality \( \frac{2}{\sqrt{x}} < M \) had to be manipulated to find an appropriate N that substantiates the limit's validity. When solving such inequalities, it is essential to recognize how operations affect their structure - for instance, squaring both sides, dividing by a negative number, or taking the reciprocal. Mastery of inequalities is a policy tool in the study of calculus, as they frequently appear in limit proofs, optimization problems, and differential equations.