Question

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Use the \(\varepsilon-\delta\) definition of infinite limits to prove that \(\lim _{x \rightarrow 3^{+}} \frac{1}{x-3}=\infty\)

Short answer

The statement is true. Using the \(\varepsilon-\delta\) definition, it can be proved that \( \lim_{x \rightarrow 3^{+}} \frac{1}{x-3}=\infty \).

Step-by-step solution

Specify the Assumed Condition

Let's start by assuming that \(M\) is any real number.

Find the Value of \(\delta\)

To find the value of \(\delta\), we need to find a real number that makes \(\frac{1}{x-3} > M\). Solving the inequality \( \frac{1}{x-3} > M\) for \(x-3\), we get \(0 < x-3 < \frac{1}{M}\). Therefore, we can choose \(\delta = \frac{1}{M}\).

Verify the Condition

Now we have to check that if \(0 < x - 3 < \delta\), then \(\frac{1}{x-3} > M\). When \(0 < x - 3 < \frac{1}{M}\), multiplying all parts of the inequality by \(M(x-3)\), we get \(0 < M(x - 3) < 1\). Since \(M(x - 3)\) is the same as \(\frac{M}{x-3}\), then it must be that \(\frac{1}{x-3} > M\). This confirms the \(\varepsilon-\delta\) definition, and hence the limit is indeed \( \infty \).

Key concepts

Infinite Limits

When we talk about infinite limits in calculus, we are examining what happens to a function as the inputs approach a specific value, but the output grows without bound. In simpler terms, as we plug numbers closer and closer to a certain point into our function, the results get larger and larger, potentially skyrocketing towards infinity.

An example of this is the function \(f(x) = \frac{1}{x-a}\), which exhibits an infinite limit as \(x\) approaches \(a\) from either side. As \(x\) gets closer to \(a\), the denominator becomes very small, and hence the value of the function becomes very large.

The \(\varepsilon-\delta\) approach provides a rigorous way of proving that a function heads towards infinity as \(x\) approaches a particular value. It states that for any large number \(M\), there's a tiny interval around \(a\) such that whenever \(x\) is in this interval (but not equal to \(a\)), the function's value exceeds \(M\). This is the formal definition used to prove infinite limits in the context of calculus.

Real Number Inequalities

Real number inequalities are statements about the relative size or order of two numbers. They're a fundamental component of calculus, especially when dealing with limits and continuity. Inequalities are used to express the range of values a number can take and still satisfy some condition.

In the context of infinite limits and the \(\varepsilon-\delta\) definition, inequalities help us define the precise conditions that values near a limit point must meet. For instance, if we want to show that \(\frac{1}{x-3} > M\) when \(x\) is close to 3 but greater than 3, we must find a range for \(x\) that makes this inequality true. This involves manipulating the inequality to isolate \(x\) and determine how close to 3 it needs to be to make the function's value exceed \(M\). Understanding how to solve and interpret these inequalities is crucial for proving statements about limits in calculus.

Calculus Proofs

Calculus proofs are logical arguments that confirm the validity of a mathematical statement within the realm of calculus. They require a clear understanding of definitions, theorems, and the tools used to analyze functions and their behaviors. Proofs in calculus often involve showing that something happens under every possible circumstance within certain constraints.

The \(\varepsilon-\delta\) definition of a limit is a cornerstone of such proofs, providing a concrete method to demonstrate the behavior of a function at a specific point. To show that the limit of \(\frac{1}{x-3}\) as \(x\) approaches 3 from the right is infinity, we must construct a proof that, no matter how large we choose a number \(M\), we can find a \(\delta\) defining an interval around 3 such that the function's value exceeds \(M\) whenever \(x\) is within that interval.

This step-by-step process not only proves the specific limit but also teaches the methods that underpin much of calculus. Mastery of these proof techniques is essential for any student seeking to understand the deeper concepts of calculus.