Problem 48 - 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 has no single continuous extension at the origin but can be extended from the right with value \( \ln(10) \) and from the left with value \(-\ln(10)\).

Step 1: Analyze the Function

The given function is \( f(x) = \frac{10^{|x|} - 1}{x} \). We will first analyze the behavior of this function as \( x \) approaches zero from both positive and negative directions.

Step 2: Approach to the Origin from the Right

Let's consider the limit as \( x \) approaches 0 from the positive side. Thus, we consider \( x \to 0^+ \). The function simplifies to \( f(x) = \frac{10^x - 1}{x} \). As \( x \to 0^+ \), \( 10^x \) approaches 1, but we need to compute the limit using L'Hôpital's Rule for indeterminate form \( \frac{0}{0} \). Both the numerator and denominator approach zero, hence apply L'Hôpital's Rule:\[\lim_{x \to 0^+} \frac{\mathrm{d}(10^x - 1)/\mathrm{d}x}{\mathrm{d}x} = \lim_{x \to 0^+} \frac{10^x \ln(10)}{1} = \ln(10).\]

Step 3: Approach to the Origin from the Left

Now, consider the limit as \( x \) approaches 0 from the negative side, \( x \to 0^- \). For the absolute value, \( 10^{|x|} = 10^{-x} \), and hence the function becomes \( \frac{10^{-x} - 1}{x} \). As \( x \to 0^- \), \( 10^{-x} \) also approaches 1, and again use L'Hôpital's Rule:\[\lim_{x \to 0^-} \frac{\mathrm{d}(10^{-x} - 1)/\mathrm{d}x}{\mathrm{d}x} = \lim_{x \to 0^-} \frac{-10^{-x} \ln(10)}{1} = - \ln(10).\]

Step 4: Determine Continuity and Extensions

The limits as \( x \to 0^+ \) and \( x \to 0^- \) are different \((\ln(10) \) and \(-\ln(10))\), which indicates there is no single continuous extension of the function \( f(x) \) at the origin. However, we have potential extensions from the right (\( f(0^+) = \ln(10) \)) and from the left (\( f(0^-) = -\ln(10) \)).

Key Concepts

L'Hôpital's Rule

In calculus, we often encounter indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). L'Hôpital's Rule is a valuable tool designed to help resolve these indecisive results. It provides a reliable method to evaluate limits that might initially seem impossible to solve.
L'Hôpital's Rule states that if you have a function limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \), and both \( f(x) \) and \( g(x) \) approach zero or infinity as \( x \) approaches \( c \), then:

• You can differentiate the numerator \( f'(x) \) and the denominator \( g'(x) \).
• Find the new limit \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \).
• If this new limit exists, it equals the original limit.

In the exercise, we see a function that produces the form \( \frac{0}{0} \) as \( x \to 0 \). By differentiating the parts of the function leverage L'Hôpital's Rule to successfully find the limit from the right and the left, even when the conventional approaches seem tricky.
This technique shines a light on otherwise obscured problems, making complex limits more accessible to evaluate.

Limit from the Right

Understanding a limit from the right means considering values of \( x \) that approach a point from the positive side (right side) on the number line. In mathematical symbolism, it is denoted as \( \lim_{x \to c^+} f(x) \).
In practical terms, as \( x \) gets closer to \( c \) from the right, we're observing what happens to \( f(x) \). If \( f(x) \) approaches a specific number, we say the limit from the right exists and equals that number. Governing this understanding, L'Hôpital's Rule can also be applied if the limit faces indeterminate forms, as it does in the exercise provided.
For the function \( \frac{10^{|x|} - 1}{x} \) as \( x \to 0^+ \), it simplifies to \( \frac{10^x - 1}{x} \). By applying L'Hôpital's Rule, the exercise shows that the expression converges to \( \ln(10) \).
Thus, we interpret that the function's behavior is approaching this specific value as it nears zero from the right. This analysis helps us understand that though the function might not be continuous at zero, a right-side limit helps us speculate and possibly assign a valuable extension from this side.

Limit from the Left

When calculating a limit from the left, we look at how \( x \) approaches a certain value from the negative side (left side) on the number line. It is expressed as \( \lim_{x \to c^-} f(x) \), meaning we are interested in the behavior of \( f(x) \) as \( x \) gets closer to \( c \) from this side. If it converges to a numerical value, this left-hand limit is well-defined and useful for analyses.
In the exercise example, we have the function \( \frac{10^{|x|} - 1}{x} \) as \( x \to 0^- \), translating to \( \frac{10^{-x} - 1}{x} \). By differentiating and utilizing L'Hôpital's Rule, it is determined that the expression simplifies to \( -\ln(10) \) when approaching zero from the left.
This approach exposes the disparity in behavior when comparing limits from the right and from the left for this particular function. While each side results in different limit values, \(-\ln(10)\) becomes a pivotal value to understand the limit on the left side. This concept is crucial for mathematical analysis and provides significant insights into potential function extensions from one side of the number line.