Problem 47 - Thomas Calculus

Question

In Exercises \(47-50,\) graph the function \(f\) to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at \(x=0\) .If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? $$ f(x)=\frac{10^{x}-1}{x} $$

Step-by-Step Solution

Answer

The function can be continuously extended at \( x=0 \) with value \( \ln 10 \).

Step 1: Identify the Function Behavior

We need to analyze the given function \( f(x) = \frac{10^x - 1}{x} \). The function is undefined at \( x = 0 \) since it results in a \( \frac{0}{0} \) indeterminate form. To check for continuity at the origin, we need to explore the function's limit as \( x \) approaches zero.

Step 2: Find the Limit Using L'Hôpital's Rule

We apply L'Hôpital's Rule to find the limit \( \lim_{x \to 0} \frac{10^x - 1}{x} \). L'Hôpital's Rule is applicable here because the function is initially in an indeterminate form \( \frac{0}{0} \). Differentiating the numerator \( 10^x - 1 \) gives \( 10^x \ln 10 \), and the derivative of the denominator \( x \) is \( 1 \). So, \( \lim_{x \to 0} \frac{10^x \ln 10}{1} = \ln 10 \cdot 10^0 = \ln 10 \).

Step 3: Interpret the Result

The result of \( \ln 10 \) indicates that as \( x \) approaches 0, the function approaches this value. Therefore, the function \( f(x) \) appears to have a continuous extension at \( x=0 \) with \( f(0) = \ln 10 \).

Step 4: Verify Continuity from Both Sides (Optional)

To ensure that the function can be extended continuously from either side, check that the limit from positive direction, \( \lim_{x \to 0^+} \frac{10^x - 1}{x} = \ln 10 \), and from the negative direction, \( \lim_{x \to 0^-} \frac{10^x - 1}{x} = \ln 10 \), are both equal to our previously found limit. This confirms continuity at the origin from both sides.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a neat trick in calculus for handling difficult limits. Imagine you're given a function that, at a certain point, presents you with an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). They sound confusing, right? Well, L'Hôpital has got your back! This rule lets you take the derivative of the top part of your fraction (the numerator) and the bottom part (the denominator), and then re-evaluate the limit. This method helps simplify those otherwise tricky scenarios.
For our function \( f(x) = \frac{10^x - 1}{x} \), as \( x \) approaches zero, the expression \( \frac{0}{0} \) pops up. That's where L'Hôpital's Rule becomes super useful. By differentiating the numerator to get \( 10^x \ln 10 \) and the denominator to just 1, the limit resolves to \( \ln 10 \). So, this rule acts like a magic wand, helping us find the limit even when it initially seems undefined.

Indeterminate Form

An indeterminate form is a kind of mathematical conundrum that pops up when limits lead to expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or even \( 0 \cdot \infty \). These forms don't have a clear value and require additional work to figure out what's going on. It's like watching a movie that stops at the most critical scene—you've got to dig deeper to understand the ending.
In the case of our function \( \frac{10^x - 1}{x} \), plugging \( x = 0 \) gives us \( \frac{0}{0} \), a classic indeterminate form. Simply put, the top and bottom both go to zero, making it tricky to know where the function's heading. With tools like L'Hôpital's Rule, however, we can break through the confusion and uncover the hidden value of the limit at that tricky point.

Function Limit Analysis

Function limit analysis is like putting on detective glasses to determine what a function is doing as it gets closer to a specific point. It helps you figure out the value a function approaches, even if it's not defined there. This concept is crucial in defining continuity and understanding behavior around tricky spots.
For the function \( f(x) = \frac{10^x - 1}{x} \), limit analysis is used to assess what's happening as \( x \) nears zero. By examining \( \lim_{x \to 0} \), we can see where the function is headed, which plays a key role in determining if we can extend it continuously at that point. The analysis concludes that the function approaches \( \ln 10 \) as \( x \) goes to zero, suggesting that \( f(x) \) can indeed be smoothly patched up, or extended, at the origin.
Understanding function limits doesn't just help with this one problem—it equips you with a powerful tool for dealing with continuity and other limit-based calculus puzzles.