Question

Infinite Limits In Exercises 73 and \(74,\) use the \(\varepsilon-\delta\) definition of infinite limits to prove the statement. $$ \lim _{x \rightarrow 5^{-}} \frac{1}{x-5}=-\infty $$

Short answer

Using the epsilon-delta definition for infinite limits, we have demonstrated that for every \(N < 0\), we can find a delta such that if the absolute difference between x and 5 is less than delta, then the function \( \frac{1}{x - 5} \) is less than \(N\). Therefore, the limit of the function \( \frac{1}{x - 5} \) as \(x\) approaches 5 from the left hand side is indeed negative infinity.

Step-by-step solution

Understand the Limit Definition for Negative Infinity

The \(\varepsilon-\delta\) definition of limit that goes to negative infinity states that \(\lim _{x \rightarrow c} f(x) = -\infty\) if for every real number \(N < 0\), there exists a \(\delta > 0\) such that if \(0<|x-c|<\delta\), then \(f(x) < N\). Here, \(f(x) = \frac{1}{x-5}\) and \(c = 5\). So we need to show that given any \(N < 0\), we can find some \(\delta > 0\) that makes \(\frac{1}{x-5} < N\) when \(0<|x-5|<\delta\).

Find the Value for Delta

The objective is to find \(\delta > 0\) such that whenever \(0 < |x - 5| < \delta\), then \(f(x) < N\). Since \(N < 0\), there must be a number \(-\frac{1}{N} > 5\). Let's set \(\delta = \frac{1}{N} - 5\). If \(0 < |x-5| < \delta\), since \(x\) is approaching \(5\) from left \(x < 5\), we can rewrite it as \(-\delta < x - 5 < 0\), then multiply with \(-1\) to get \(0 < 5 - x < \delta\) and \(\frac{1}{5-x} > \frac{1}{\delta} = -N\). Hence \(f(x) = \frac{1}{x-5} < N\).

Make a Conclusion

Therefore, we have proved that for every \(N < 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 5| < \delta\), then \(f(x) = \frac{1}{x - 5} < N\). This means \(\lim _{x \rightarrow 5^{-}} \frac{1}{x-5} = -\infty\) with the \(\varepsilon-\delta\) definition of infinite limits. This completes the exercise's requirement.

Key concepts

Epsilon-Delta Definition

The Epsilon-Delta definition is a fundamental concept in calculus. It helps us formally define what it means for a limit to exist. When we say a function approaches a certain limit as it nears a point, the Epsilon-Delta definition gives us the detailed criteria. In simpler terms, it assures us how close we need to be to a particular point (delta, \(\delta\)) for the function value to be within a certain accuracy (epsilon, \(\varepsilon\)).

In the context of infinite limits, things are a bit different. Instead of \(\varepsilon\), we often use a parameter like \(N\) which represents negative infinity when working with limits approaching \(-\infty\). Here's the essence of the definition:

• If \(\lim _{x \rightarrow c} f(x) = -\infty\), for every real number \(N < 0\), there must be a \(\delta > 0\) such that whenever \(0<|x-c|<\delta\), the function \(f(x) < N\).

This structured approach allows mathematicians to rigorously determine behavior as x approaches specific values.

Negative Infinity

The concept of negative infinity in limits is rather intriguing. It signifies that as we get closer to a certain point, the values of the function decrease without bound. This behavior means the function is heading towards values that are infinitely negative. Such scenarios are essential for understanding the behavior of certain types of functions, particularly rational functions.

In the problem given, for example, as \(x\) approaches \(5\) from the left, the function \(\frac{1}{x-5}\) moves downward towards \(-\infty\). This happens because as \(x\) gets closer to 5, the value \(x-5\) becomes a very small negative number, making \(\frac{1}{x-5}\) a very large negative number.

• Negative infinity helps us understand where graphs are heading.
• It’s crucial for understanding the end behavior of functions.

Understanding and using negative infinity is vital for correctly analyzing and solving calculus problems.

Limit Proof

Proving limits, especially those that tend to infinity, involves a methodical process. The epsilon-delta or, in this case, \(N-\delta\) proof structure is key. It's not just about finding an answer, but ensuring that the behavior of functions is predictable and measurable as the input approaches a specific value.
Here's a summary of the proof process for negative infinity:

• Identify the function and the point where the limit is approached.
• Find a condition using \(\delta\) that ensures the function meets the requirement for reaching \(-\infty\). In our exercise, this was \(\delta = \frac{1}{N} - 5\).
• Show that for any chosen negative \(N\), the function will always be less than \(N\) within the delta interval.

These steps guarantee that as \(x\) approaches the value on the graph, the function behaves in the manner predicted: it tends towards minus infinity.

Calculus Exercises

Working through calculus exercises like proving infinite limits builds your skills and intuition for mathematical concepts. Practicing these problems helps break down the complex moves in calculus into more manageable pieces. It's this regular exposure and repetition that builds proficiency in handling complex mathematical theories and operations.
When solving a calculus problem:

• Understanding the problem statement is crucial. Notice key points and behavior it expresses.
• Translate those behaviors into mathematical expressions or definitions, as with the Epsilon-Delta definition here.
• Execute a step-by-step approach to prove the statement or solve the problem. Fill in the logic between problem statement and solution.

With each exercise, your familiarity with calculus's foundational building blocks strengthens. These exercises are perfect practice not just for exams, but for grasping calculus's real-world applications.