Find lim_(h rarr0)(2tan((pi)/(3)+h)-2tan((pi)/(3)))/(h). Choose 1 answer: (A) 1 (B) 2 (C) 8 (D) The limit doesn't exist

Question

Find  lim_(h rarr0)(2tan((pi)/(3)+h)-2tan((pi)/(3)))/(h). Choose 1 answer: (A) 1 (B) 2 (C) 8 (D) The limit doesn't exist

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Identify Problem: Identify the limit problem and recognize it as a derivative problem at a specific point.Use Derivative Definition: Use the definition of the derivative f'(a) = lim_(h -> 0)(f(a+h)-f(a))/h for the function f(x) = 2tan(x) at x = pi/3.Plug in Values: Plug in the values into the derivative formula: f'(pi/3) = lim_(h -> 0)(2tan(pi/3+h)-2tan(pi/3))/h.Simplify Expression: Simplify the expression inside the limit: 2tan(pi/3+h)-2tan(pi/3) = 2(tan(pi/3+h)-tan(pi/3)).Apply Tangent Formula: Apply the tangent addition formula: tan(a+b) = (tan(a)+tan(b))/(1-tan(a)tan(b)), where a = pi/3 and b = h.Substitute Values: Substitute tan(pi/3) = sqrt(3) into the formula: tan(pi/3+h) = (sqrt(3)+tan(h))/(1-sqrt(3)tan(h)).Plug Expression Back: Plug the expression for tan(pi/3+h) back into the limit: lim_(h -> 0)(2((sqrt(3)+tan(h))/(1-sqrt(3)tan(h))-sqrt(3)))/h.Simplify Inside Limit: Simplify the expression inside the limit: (2(sqrt(3)+tan(h))/(1-sqrt(3)tan(h))-2sqrt(3))/h = 2(tan(h)/(1-sqrt(3)tan(h)))/h.Divide by h: Divide by h inside the limit: lim_(h -> 0)(2tan(h)/((1-sqrt(3)tan(h))h)).Recognize Approach: Recognize that as h approaches 0, tan(h)/h approaches 1.Substitute in Limit: Substitute tan(h)/h with 1 in the limit: lim_(h -> 0)(2/(1-sqrt(3)tan(h))).Evaluate Limit: Evaluate the limit as h approaches 0: 2/(1-sqrt(3)tan(0)) = 2/(1-0) = 2.Conclude Solution: Conclude that the value of the limit is 2, which corresponds to answer choice (B).

Find $\lim _{h \rightarrow 0} \frac{2 \tan \left(\frac{\pi}{3}+h\right)-2 \tan \left(\frac{\pi}{3}\right)}{h}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $1$ $\newline$ (B) $2$ $\newline$ (C) $8$ $\newline$ (D) The limit doesn't exist

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