Question

Find the limit as x approaches 0 from the right of xᵃ. ln x where a is a positive constant.

I've gone ahead and assumed it requires l'hopitals rule - meaning product rule. So I've got axᵃ⁻¹ln x+ xᵃ/x. Where to from here?
a) 0
b) 1
c) [infinity]
d) Does not exist

Answer 1

To find the limit as x approaches 0 from the right of xᵃ ln x, you can use L'Hopital's Rule. Differentiating the numerator and denominator, we get axᵃ⁻¹ln x + xᵃ/x. As x approaches 0 from the right, both terms involving x approach 0 and the limit becomes 0.

Step-by-step explanation:

To find the limit as x approaches 0 from the right of xᵃ ln x, we can use L'Hopital's Rule since it involves an indeterminate form (0 * ∞).

Applying L'Hopital's Rule, we differentiate the numerator and the denominator. The derivative of xᵃ is a xⁱ⋅ln x + xᵃ/x. The derivative of ln x is 1/x.

Substituting these derivatives back into the expression, we get a xⁱ⋅ln x + xᵃ/x. As x approaches 0 from the right, both terms involving x approach 0 and the limit becomes 0.