Problem 18 - The Calculus with Analytic Geometry

Question

Prove that \(\lim _{x \rightarrow+\infty}\left(x^{2}-4\right)=+\infty\) by showing that for any \(N>0\) there exists an \(M>0\) such that \(\left(x^{2}-4\right)>N\) whenever \(x>M\)

Step-by-Step Solution

Answer

\(M = \sqrt{N + 4}\)

Step 1 - State the definition of the limit to infinity

To prove that \(\lim_{x \rightarrow +\infty} \(x^{2} - 4\) = +\infty\), we need to show that for any given \(N > 0\), there exists an \(M > 0\) such that \(x^{2} - 4 > N\) whenever \(x > M\).

Step 2 - Set up the inequality

Start with the inequality \(x^{2} - 4 > N\). This needs to be true whenever \(x > M\).

Step 3 - Rearrange the inequality

Rearrange \(x^{2} - 4 > N\) to solve for \(x\):\(\ x^{2} > N + 4 \).

Step 4 - Take the square root

Take the square root of both sides to isolate \(x\):\(\ x > \sqrt{N + 4} \).

Step 5 - Define the value of M

Let \(M = \sqrt{N + 4} \). This ensures that for any \(x > M\), \(x^{2} - 4 > N\).

Step 6 - Conclusion

Since for any \(N > 0\), we can find \(M = \sqrt{N + 4} \) such that \(x^{2} - 4 > N\) whenever \(x > M\), we have shown that \(\lim_{x \rightarrow +\infty} \(x^{2} - 4\) = +\infty\).

Key Concepts

inequalities

Inequalities are used to compare two values or expressions. When we talk about inequalities, we generally use symbols like > (greater than) and < (less than). They help us understand how different quantities relate to each other.

An important point here is rearranging inequalities. For example, given the inequality:\( x^{2} - 4 > N \), we want to find the range of values for x that make the inequality true. This often involves steps like adding or subtracting terms and taking square roots.

In our example, we rearranged \( x^{2} - 4 > N \) to \( x^{2} > N + 4 \), and then solved it by taking the square root of both sides to get:\( x > \sqrt{N + 4} \). Rearranging and simplifying inequalities help isolate the variable we're interested in, making it easier to draw conclusions about possible values.

limit definition

A limit is a fundamental concept in calculus. When we say \( \lim_{x \to +\infty} \), it means we are looking at the behavior of a function as the variable x grows very large.

In our given problem, we want to show that \( \lim_{x \to +\infty} (x^{2} - 4) = +\infty \). This means as x increases to very large values, the expression \( x^{2} - 4 \) also grows without bound.

To rigorously prove this using the limit definition, we need to demonstrate that for any arbitrarily large number N, there exists another number M such that if x is larger than M, then the expression \( x^{2} - 4 \) is greater than N. Essentially, we are creating a relationship between our large output (N) and our input (x) that confirms the growth pattern of the function.

This approach ensures that no matter how large you choose N to be, there will always be a sufficiently large x (greater than M) for which the function's value exceeds N, proving our limit assertion.

square root

Taking the square root of a number is a fundamental mathematical operation. The square root of a number y, denoted as \( \sqrt{y} \), is a value z such that \( z^{2} = y \).

In solving the inequality \( x^{2} - 4 > N \), we needed to isolate x. Once we rearranged it to \( x^{2} > N + 4 \), we took the square root of both sides to find that \( x > \sqrt{N + 4} \).

Taking the square root helps simplify equations involving squares because it reverses the squaring process. However, it's important to consider the properties of square roots in context:

• Square roots of positive numbers have two solutions: a positive and a negative root. For instance, \( \sqrt{9} \) is both 3 and -3 because \( 3^{2} = 9 \) and \( -3^{2} = 9 \).
• In inequality problems, like ours, we usually focus on the principal (positive) square root when considering limits to positive infinity.

By understanding and correctly applying these principles, you can confidently manipulate and solve square root expressions in calculus problems.