Question

We write \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M\) there is a corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x^{2}+x}{x}=\infty$$

Short answer

Question: Prove that the limit of the function \(f(x) = \frac{x^2 + x}{x}\) goes to infinity as x approaches infinity. Answer: To prove that the given limit goes to infinity, the function has been simplified to \(f(x) = x + 1\). After establishing the inequality \(x+1>M\), we find that for any positive M, there exists an N such that \(f(x) > M\) whenever \(x > N\). Therefore, we can conclude that \(\lim_{x \rightarrow \infty} \frac{x^2 + x}{x} = \infty\).

Step-by-step solution

Simplify the function

First, we'll simplify the function f(x) by dividing x^2 and x by x. This gives: $$f(x) = \frac{x^2 + x}{x} = x + 1$$

Set up inequality for M

Now we must show that \(f(x) > M\) for \(x > N\). We can start by setting up the inequality: $$x + 1 > M$$

Solve for x

Next, we'll solve this inequality by isolating x: $$x > M - 1$$

Establish the relationship between M and N

Now, we know that \(x > N\) by definition, and from the previous step, we find \(x > M - 1\). This means that we must have: $$N > M - 1$$

Solve for N

Now, we simply add 1 to both sides of the inequality to solve for N: $$N = M$$

Conclusion

We have shown that for any positive M, there exists a N such that \(f(x) > M\) whenever \(x > N\). Therefore, by definition, we can conclude that: $$\lim_{x \rightarrow \infty} \frac{x^2 + x}{x} = \infty$$

Key concepts

Proof Strategy for Limits

Understanding how to prove that the limit of a function as x approaches infinity is infinite involves a clear strategy based on a formal definition. This strategy is essential for showing with rigor that as x grows without bound, the function increases indefinitely.

In proving that \( \lim_{x \rightarrow \infty} f(x) = \infty \), one aims to show that no matter how large a positive number \(M\) we choose, there is always a point \(N\) beyond which all function values are greater than \(M\). The selection of \(M\) is arbitrary, representing any height we set, no matter how high. Now, to establish the proof, we must:

• Apply the given function to observe its behavior as x approaches infinity.
• Set up an inequality involving \(M\) that represents the condition that \(f(x) > M\).
• Solve this inequality for x to determine how large x needs to be to ensure the inequality holds.
• State \(N\) in terms of \(M\), forming the explicit relationship required by the definition.

For the specific exercise, simplifying the function \(f(x) = \frac{x^2 + x}{x}\) to \(x + 1\) and then manipulating the inequality \(x + 1 > M\), gives us the relationship \(N = M\) to complete the proof.

Infinite Limits

When discussing limits at infinity, the term 'infinite limits' speaks directly to the behavior of functions as their inputs grow without bound. Specifically, if a function grows larger than any predetermined value as the input increases, we say the limit of that function as x tends toward infinity is infinite.

Symbolically, this is represented as \(\lim_{x \rightarrow \infty} f(x) = \infty\). It signifies that no matter how far along the x-axis we move, the function value will surpass any fixed level we might choose. This concept is not about reaching an ultimate value but about understanding behavior—there's no bounding the height that the function's value can achieve.

In practice, we might encounter functions that simplistically indicate a trend towards infinity, such as \(f(x) = x + 1\) in our exercise. No upper cap exists for this increasing linear function, and therefore we confidently designate its limit at infinity as infinite.

Limit Properties

Limit properties formalize fundamental relationships and operations for limits and provide the underlying framework to manipulate and evaluate limits systematically. Some of the essential limit properties involve sums, differences, products, and quotients of functions.

Using these properties allows us to break down complex limits into more manageable parts. For instance, when presented with a complex rational function like \(f(x) = \frac{x^2 + x}{x}\), we can apply the property that the limit of a sum is the sum of the limits (if those limits exist) and the property that the limit of a function times a scalar is the scalar times the limit of the function, to simplify this to \(x + 1\). These simplifications are powerful tools because they transform tougher limit evaluations into simpler problems we can solve with basic algebra. The exercise shows such an application, starting with a fraction and simplifying down to a linear expression whose limit at infinity can be more straightforwardly determined as infinite.