Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.

Verified step by step guidance

1. First, recognize that the function is given by \(f(x) = x^{p} \ln x\), and we are interested in the behavior as \(x \to 0^{+}\). Since \(\ln x\) tends to \(-\infty\) as \(x \to 0^{+}\), the product's behavior depends on the power \(p\) of \(x\).
2. Rewrite the function to analyze the limit: \(f(x) = x^{p} \ln x = \frac{\ln x}{x^{-p}}\). As \(x \to 0^{+}\), \(x^{-p} \to +\infty\), so this is an indeterminate form of type \(\frac{-\infty}{+\infty}\), which suggests using L'Hôpital's Rule.
3. Apply L'Hôpital's Rule by setting \(g(x) = \ln x\) and \(h(x) = x^{-p}\). Compute the derivatives: \(g'(x) = \frac{1}{x}\) and \(h'(x) = -p x^{-p-1}\). Then, the limit becomes \(\lim_{x \to 0^{+}} \frac{g'(x)}{h'(x)} = \lim_{x \to 0^{+}} \frac{\frac{1}{x}}{-p x^{-p-1}}\).
4. Simplify the expression: \(\frac{\frac{1}{x}}{-p x^{-p-1}} = \frac{1}{x} \cdot \frac{1}{-p} \cdot x^{p+1} = -\frac{1}{p} x^{p}\). As \(x \to 0^{+}\), \(x^{p} \to 0\), so the limit is \(0\). This shows that \(\lim_{x \to 0^{+}} x^{p} \ln x = 0\) for all \(p > 0\).
5. To understand how the graphs differ for \(p = \frac{1}{2}, 1,\) and \(2\), note that the factor \(x^{p}\) approaches zero at different rates. For smaller \(p\), \(x^{p}\) approaches zero more slowly, so the negative logarithmic term dominates more, causing the graph to dip more sharply near zero. For larger \(p\), \(x^{p}\) goes to zero faster, so the product approaches zero more gently. Sketching these will show the curve for \(p=\frac{1}{2}\) dipping down more steeply near zero, while for \(p=2\) it approaches zero more smoothly.

Key Concepts

Behavior of Functions Near Zero

Understanding how functions behave as x approaches zero from the right (x → 0⁺) is crucial. This involves analyzing limits and determining whether the function approaches zero, infinity, or a finite value, which helps in sketching accurate graphs near the origin.

Properties of the Natural Logarithm Function

The natural logarithm ln(x) tends to negative infinity as x approaches zero from the right. This behavior significantly influences the product xᵖ ln(x), especially since ln(x) dominates near zero, affecting the overall shape and limit of the function.

Effect of the Power p in xᵖ

The exponent p controls how quickly xᵖ approaches zero as x → 0⁺. Different values of p (1/2, 1, 2) change the rate at which the product xᵖ ln(x) tends to zero or diverges, impacting the graph's slope and curvature near the origin.