Problem

Find the following limit. \[ \lim _{x \rightarrow 0^{+}}(2 x \cot (\pi-x)) \]

Solution

Step 1 :The given limit is \(\lim _{x \rightarrow 0^{+}}(2 x \cot (\pi-x))\).

Step 2 :As x approaches 0 from the right, the limit is of the form 0/0. Therefore, we can use L'Hopital's rule to solve this limit.

Step 3 :L'Hopital's rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

Step 4 :We need to find the derivative of the numerator and the denominator. The derivative of 2x is 2. The derivative of cot(x) is -csc^2(x), and the derivative of π-x is -1. Therefore, the derivative of cot(π-x) is csc^2(π-x).

Step 5 :After finding the derivatives, we substitute them back into the limit and solve.

Step 6 :The limit of the function as x approaches 0 from the right is 0.

Step 7 :Final Answer: The limit is \(\boxed{0}\).

From Solvely APP
Source: https://solvelyapp.com/problems/24968/

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