Question

Graph the function \(f_{k}(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.5,\) and 0.1 on \([0,10] .\) Find \(\lim _{k \rightarrow 0^{+}} f_{k}(x) .\)

Short answer

The graph of the function shows different shapes for different values of \(k\). The limit of the function as \(k\) approaches 0 from the positive side is \(-\infty\).

Step-by-step solution

Understanding the Function

Start by plugging in the various given values for \(k\) into \(f_{k}(x)=\frac{x^{k}-1}{k}\). Do this for \(k = 1\), \(0.5\), and \(0.1\). Now, graph this on the interval \([0, 10]\).

Graphing the Function

The function will look different at each value of \(k\) because \(k\) affects the rate at which \(x\) raises to the power of \(k\). Pay attention while graphing this as the shape and nature of the graph are important here.

Finding the Limit

In order to find the limit of this function as \(k\) approaches 0 from the right, we need to make use of the fact that \(\lim_{k \rightarrow 0^{+}} f_{k}(x) = f(x)\) where \(f(x)\) is the function with \(k\) replaced by 0.

Evaluating the Limit

The function becomes \(f(x) = \frac{x^0-1}{0}\) which simplifies down to \(-\infty\) due to division by zero. Remember, the result will not be the same if the limit approaches from the negative side. Here the condition specified is \(k \rightarrow 0^{+}\), which means \(k\) is approaching 0 from the positive side. The result is the limit as \(k\) approaches 0 from the positive side.