Problem 43 - Calculus: Early Transcendentals

Question

Prove that \( \displaystyle \lim_{x \to 0^+} \ln x = -\infty \).

Step-by-Step Solution

Answer

As \( x \to 0^+ \), \( \ln x \) approaches \(-\infty\) by choosing \( \delta = e^L \) for any \( L < 0 \).

Step 1: Understanding the Limit Definition

We want to prove that as \( x \) approaches 0 from the positive side, \( \ln x \) approaches \(-\infty\). This means that for any large negative number \( L \), there exists a \( \delta > 0 \) such that if \( 0 < x < \delta \), then \( \ln x < L \).

Step 2: Transforming the Expression

Start by considering the inequality \( \ln x < L \). By exponentiating both sides, we transform the inequality to \( x < e^L \).

Step 3: Finding \( \delta \) in Terms of \( L \)

To satisfy the inequality \( 0 < x < e^L \), we can choose \( \delta = e^L \). This ensures that if \( x < \delta \), then \( x < e^L \), satisfying our condition.

Step 4: Concluding the Proof

Since for any large negative number \( L \), we can find \( \delta = e^L \) such that \( 0 < x < \delta \) implies \( \ln x < L \), we have shown that \( \lim_{x \to 0^+} \ln x = -\infty \).

Key Concepts

Epsilon-Delta Definition

To understand the epsilon-delta definition of limits, let's simplify it. When we talk about limits, especially as a function approaches a particular value, the epsilon-delta definition provides a formal way to express this behavior.

In this setting,

• "Epsilon" (usually denoted as \( \epsilon \)) represents how close the function gets to a certain limit value.
• "Delta" (\( \delta \)) shows how close the input (or \( x \)) must be to a particular point to achieve that level of closeness in the function value.

For the limit of the natural logarithm as \( x \) approaches 0 from the positive side, the definition takes a specific form. We want \( \ln x \) to become arbitrarily large in the negative direction (towards \(-\infty\)) when \( x \) gets close to 0. In this case, we look for a \( \delta \) such that for any negative number \( L \), if \( 0 < x < \delta \), then \( \ln x < L \). This defines the relationship between how small \( x \) must be relative to \( 0 \) (the \( \delta \)) to control how small \( \ln x \) becomes relative to \( L \).

Through this definition, you get a sturdy understanding of limits and how they function even at extreme positions of the variable.

Natural Logarithm Behavior

Natural logarithm \( \ln x \) is a fascinating function, especially as it approaches the limits on the graph. One crucial behavior is how \( \ln x \) dips towards negative infinity as \( x \) nears zero from the right.

Here's how it works:

• The natural logarithm function is undefined for 0 and negative numbers, meaning you can't take the log of these values.
• As \( x \) decreases but remains positive, the values of \( \ln x \) will head downward towards \(-\infty\).
• This is because the logarithmic function reflects how many times we need to magnify a base (e.g., \( e \)) to reach a particular number.

As \( x \) gets closer and closer to 0, the expressivity of this behavior increases because it takes a much larger (negative) power to shrink the base to match the tiny approaching number \( x \). Hence, you'll note the seemingly deep plunge of \( \ln x \) as \( x \to 0^+ \).

Understanding this behavior is key to analyzing the limits of \( \ln x \), especially when dealing with infinite or extreme values.

Approaching Zero from the Right

In mathematics, approaching a value from the right means you're looking at values slightly bigger than that value. When \( x \to 0^+ \), \( x \) is getting very close to zero, but always positive.

Why is this important?

• It's crucial because many functions, including \( \ln x \), have different behaviors as they approach zero from the right versus the left.
• For \( \ln x \), approaching from the right ensures that \( x \) never becomes zero or negative, thus keeping the function defined.
• By examining \( x \to 0^+ \), we focus specifically on the side where the natural logarithm has meaningful values in the real number sense.

This directionality (from the right) helps us assert that \( \ln x \) plunges indefinitely below, behaving as anticipated into negative infinity, while avoiding undefined realms. By approaching from this side, we can properly set the stage for limit evaluations like \( \lim_{x \to 0^+} \ln x = -\infty \).

In summary, analyzing behavior as \( x \to 0^+ \) provides insights into the practical limits for functions that diverge or converge intensely near certain points.