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lim-x-0-1-tan-x-1-tan-x-1-x-




Question Number 194256 by horsebrand11 last updated on 01/Jul/23
    lim_(x→0)  ((((1+tan x)/(1−tan x)) −1)/x) =?
$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}+\mathrm{tan}\:\mathrm{x}}{\mathrm{1}−\mathrm{tan}\:\mathrm{x}}\:−\mathrm{1}}{\mathrm{x}}\:=? \\ $$
Answered by tri26112004 last updated on 01/Jul/23
= lim_(x→0)  ((2tan x)/(x(1−tan x)))  =lim_(x→0)  ((tan x)/x) . lim_(x→0)  (2/(1−tan x))  = 1 . (2/(1−0)) = 2
$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{tan}\:{x}}{{x}\left(\mathrm{1}−{tan}\:{x}\right)} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{tan}\:{x}}{{x}}\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}}{\mathrm{1}−{tan}\:{x}} \\ $$$$=\:\mathrm{1}\:.\:\frac{\mathrm{2}}{\mathrm{1}−\mathrm{0}}\:=\:\mathrm{2} \\ $$
Answered by cortano12 last updated on 01/Jul/23
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$$\:\:\:\cancel{\lesseqgtr} \\ $$

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