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Question Number 130869 by EDWIN88 last updated on 30/Jan/21

lim_(x→0) (((tan x)/x))^((tan x)/(x−tan x )) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{tan}\:{x}}{{x}}\right)^{\frac{\mathrm{tan}\:{x}}{{x}−\mathrm{tan}\:{x}\:}} ? \\ $$

Answered by benjo_mathlover last updated on 30/Jan/21

ln L = lim_(x→0) (((tan x)/(x−tan x)))ln (((tan x)/x)) ln L= lim_(x→0) ((ln (((tan x)/x)))/(((x/(tan x))−1))) ; let ((tan x)/x) = t ln L = lim_(t→1) ((ln t)/((1/t)−1)) = lim_(t→1) ((1/t)/(−(1/t^2 ))) = −1 L = e^(−1)

$$\:\mathrm{ln}\:\mathrm{L}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{x}−\mathrm{tan}\:\mathrm{x}}\right)\mathrm{ln}\:\left(\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{x}}\right) \\ $$$$\:\mathrm{ln}\:\mathrm{L}=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{x}}\right)}{\left(\frac{\mathrm{x}}{\mathrm{tan}\:\mathrm{x}}−\mathrm{1}\right)}\:;\:\mathrm{let}\:\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{x}}\:=\:\mathrm{t} \\ $$$$\:\mathrm{ln}\:\mathrm{L}\:=\:\underset{\mathrm{t}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\mathrm{t}}{\frac{\mathrm{1}}{\mathrm{t}}−\mathrm{1}}\:=\:\underset{\mathrm{t}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{t}}}{−\frac{\mathrm{1}}{\mathrm{t}^{\mathrm{2}} }}\:=\:−\mathrm{1} \\ $$$$\:\mathrm{L}\:=\:\mathrm{e}^{−\mathrm{1}} \: \\ $$

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