Problem 19 - Calculus for Biology and Medicine

Question

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 3^{+}} \frac{1}{3-x}=-\infty $$

Step-by-Step Solution

Answer

As \( x \rightarrow 3^{+} \), \( \frac{1}{3-x} \) decreases without bound, confirming \( \lim_{x \rightarrow 3^{+}} \frac{1}{3-x} = -\infty \).

Step 1: Understanding the Limit Notation

We want to show that as \( x \) approaches 3 from the right, the limit of \( \frac{1}{3-x} \) is \(-\infty\). This means that for every large positive number \( M \), there exists a \( \delta > 0 \) such that whenever \( 3 < x < 3 + \delta \), \( \frac{1}{3-x} < -M \).

Step 2: Expressing \( \frac{1}{3-x} < -M \)

Start by rewriting the inequality \( \frac{1}{3-x} < -M \) which implies \( 3-x < \frac{-1}{M} \). This shows that \( x-3 > \frac{1}{M} \) and due to the absolute value in the formal definition of limits, we want \( 0 < x-3 < \delta \).

Step 3: Finding \( \delta \)

We need \( \frac{1}{M} < x-3 < \delta \). Therefore, choose \( \delta = \frac{1}{M} \). This ensures that whenever \( x \) is in the interval \( (3, 3 + \delta) \), the inequality \( \frac{1}{3-x} < -M \) holds true.

Step 4: Formal Proof

For every \( M > 0 \), select \( \delta = \frac{1}{M} \). Assume \( 0 < x - 3 < \delta \). Then \( x-3 < \frac{1}{M} \) or equivalently \( 3-x > \frac{-1}{M} \), which implies \( \frac{1}{3-x} < -M \). Hence, the limit \( \lim_{x \rightarrow 3^{+}} \frac{1}{3-x} = -\infty \) is proven.

Key Concepts

Limit Proof

Proving limits can seem daunting, but breaking down the process helps simplify things. In our example, the goal is to show that as \( x \) approaches 3 from the right, the expression \( \frac{1}{3-x} \) tends towards \(-\infty\). This indicates that, for every large positive number \( M \), there must be a small positive number \( \delta \) such that whenever \( x \) is within the interval \( (3, 3 + \delta) \), the expression \( \frac{1}{3-x} \) becomes smaller than \(-M\). To prove this, we convert the inequality \( \frac{1}{3-x} < -M \) into another form: \( 3-x < \frac{-1}{M} \). By reorganizing terms, we find \( x-3 > \frac{1}{M} \), and to fit the limits definition, we need:

• \( 0 < x-3 < \delta \)
• Ensure \( \delta = \frac{1}{M} \)

This choice of \( \delta \) ensures that whenever \( x \) is slightly greater than 3, the function's value becomes overwhelmingly negative. In such proofs, selecting \( \delta \) based on \( M \) helps us bring rigorous precision to the expectation that limit values dictate.

Formal Definition of Limits

At the heart of calculus limits is a precise definition that ensures mathematics works the way we expect it to. For the function limit, the statement \( \lim_{x \to a} f(x) = L \) relies on this formal definition. It means that for every small positive number \( \epsilon \), there exists a small positive number \( \delta \) such that whenever \( x \) is within \( 0 < |x-a| < \delta \), the function \( f(x) \) is within \( |f(x) - L| < \epsilon \) of \( L \). This precise definition allows us to prove the behavior of functions near a point without relying on visual graphs.In the problem given, as \( x \) approaches 3 from values greater than 3, the value \( \frac{1}{3-x} \) decreases to \(-\infty\). The formal definition ensures we can state this behavior using clearly defined parameters \( M \) and \( \delta \).1. Set a large \( M \) to represent a "cut-off" value that \( \frac{1}{3-x} \) must go beneath.2. Determine \( \delta \) such that if \( x \) strays from 3 within this distance, our inequality is satisfied.This structure provides a robust way to convey, validate, and understand the behavior of limits mathematically.

One-Sided Limit

One-sided limits focus on what happens to a function as values approach a specific number from one direction: either the left or the right. This is crucial in understanding behaviors like discontinuous points or vertical asymptotes, where differing behaviors occur on either side of a specific point.In our example, \( \lim_{x \rightarrow 3^{+}} \frac{1}{3-x} = -\infty \), specifically indicates a right-hand limit, where \( x \) creeps up to 3 from values greater than 3. Such a limit takes into account the behavior of the function when approaching that point strictly from one side, highlighting any asymmetrical tendencies.This emphasis on directionality is vital because:- Different behavior might occur when approaching from different sides.- It highlights infinite discontinuities or vertical asymptotes where the limit can be \( \infty \) or \(-\infty\). Understanding one-sided limits assists in sketching accurate graphs and identifying asymmetrical behaviors in functions, allowing us to capture a complete picture of their behavior near specified points.