Final answer:
The given limit as x approaches 0⁺ of (2/x - 1/tan(x)) is indeterminate and requires the application of L'Hôpital's rule by differentiating both the numerator and denominator before taking the limit again.
Step-by-step explanation:
To Find the limit as x approaches 0⁺ of (2/x - 1/tan(x)), we need to use trigonometric identities and limits properties. The limit of 2/x as x approaches 0⁺ is infinity and the limit of 1/tan(x) can be rewritten using the identity tan(x) = sin(x)/cos(x), which gives 1/sin(x) * cos(x). However, as x approaches 0, cos(x) approaches 1 and 1/sin(x) approaches infinity. Therefore, we have a form of ∞ - ∞, which is indeterminate and requires further methods such as L'Hôpital's rule to evaluate. Applying L'Hôpital's rule by differentiating both the numerator and the denominator and then taking the limit, we may get a determinate value. This calculus technique is often used when straightforward substitution leads to indeterminate forms like 0/0 or ∞/∞. In this case, differentiating 2/x gives -2/x² and differentiating tan(x) using the quotient rule or by recognizing that tan(x) = sin(x)/cos(x) and using the product rule, we get sec²(x) (the derivative of tan(x)) in the denominator. Only then can we evaluate the limit properly.