Problem 80 - Essential Calculus

Question

Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).

Step-by-Step Solution

Answer

The limit as k approaches 0 from the positive side for the function \(f(x)=\frac{x^{k}-1}{k}\) is 0.

Step 1: Graph the Function

Use a graphing utility to plot \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1, 0.1\), and 0.01. These graphs will give a visual representation of the function for different values of k.

Step 2: Observe the Trends

From the graphs, observe how the function changes as k gets closer to zero. This will be useful in predicting the behavior of the function when calculating the limit.

Step 3: Calculate the Limit

Next, proceed to evaluate the limit \( \lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k} \). This involves calculating \(x^{k}-1\) divided by k as k gets infinitesimally close to 0 from the positive side.

Step 4: Use L'Hopital's Rule

This limit could be in the indeterminate form of \(0/0\) or \(∞/∞\) as k approaches 0, so L'Hopital's rule can be used. It states that \( \lim _{x \rightarrow a} [f(x)/g(x)] = \lim _{x \rightarrow a} [f'(x)/g'(x)] \), whenever the limit on the right side exists. Here, \(f(x) = x^{k}-1\) and \(g(x) = k\). The derivative of \(f(x)\) with respect to x is \(k*x^{k-1}\), and the derivative of \(g(x)\) is 1.

Step 5: Substitute the Limit

Now, substitute the limit \(k \rightarrow 0\) in the function \(\lim _{k \rightarrow 0^{+}} \frac{k*x^{k-1}}{1} \). The expression reduces to zero as k in the numerator becomes 0.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions on a graph. It helps students and mathematicians alike to better understand the behavior and trends of a particular function over an interval.
By plotting the function using a graphing utility, such as a graphing calculator or software like Desmos, you can analyze how different parameters affect the function. For this particular exercise, the function used is \( f(x)=\frac{x^{k}-1}{k}\).
The graphing utility allows you to see how the graph behaves for varying values of \( k \) such as 1, 0.1, and 0.01. This visual representation is useful for making conjectures about the behavior of the function, especially when predicting how it responds as \( k \) approaches zero. Understanding trends from the graph can make the process of solving calculus problems much clearer.

L'Hopital's Rule

L'Hopital's Rule is an essential tool in calculus used to evaluate limits that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
This rule simplifies complex limit calculations by differentiating the numerator and the denominator separately. It's specifically applicable when both the top and bottom parts of a fraction converge towards zero or infinity as the variable approaches a certain value.
In applying this rule to the given function, \( f(x)=x^{k}-1 \) is used as the numerator and \( g(x)=k \) as the denominator. As explained in the exercise, you take the derivative of \( f(x) \), resulting in \( k*x^{k-1} \), and the derivative of \( g(x) \) is straightforward, as it is 1.
This means the complicated limit problem becomes much easier to tackle, allowing you to substitute \( k \to 0 \) directly into \( \frac{k*x^{k-1}}{1} \). This simplification doesn't just make calculations more manageable, it often leads directly to the solution.

Indeterminate Forms

Indeterminate forms are expressions encountered in calculus that do not have a straightforward value. You often see these as limits that, at face value, could signify multiple possibilities, such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), 0/\( \infty \), or similar forms.
When confronted with an indeterminate form, it often signals the need for more sophisticated techniques, like L'Hopital's Rule, which provides a way to resolve these uncertainties by transforming the problematic part of the expression. This helps in getting a clearer numerical or symbolic value.
In this exercise, when evaluating \( \lim_{k \to 0^{+}} \frac{x^{k}-1}{k} \), as \( k \) approaches zero, it originally seems indeterminate. However, by using appropriate methods such as derivatives, you gain insight into the true limit by moving past the apparent indeterminacy.

Calculus Problem Solving

Calculus problem solving often involves a step-by-step approach to breaking down complex functions. Like with the function \( f(x) = \frac{x^{k}-1}{k} \), begin with analyzing parts of the function.
Such exercises typically include analyzing and graphing the function to understand its behavior visually. From there, explore any indeterminate forms that may arise during limit calculations. Use tools and rules like L'Hopital’s to meticulously simplify and handle challenging expressions.
When solving calculus problems like this, persistence and using proven strategies are key in avoiding common pitfalls. Always ensure each step logically follows, keeping mathematical principles in check while aiming for clarity and correctness.
Being methodical in evaluating limits not only sharpens skills but also makes grasping underlying principles of calculus much easier.