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 \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).

Short answer

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

Step-by-step solution

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.

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.

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.

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.

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.