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lim-x-0-e-x-e-x-2-x-2-




Question Number 87153 by jagoll last updated on 03/Apr/20
lim_(x→0)  ((e^x +e^(−x) −2)/x^2 )
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} −\mathrm{2}}{\mathrm{x}^{\mathrm{2}} } \\ $$
Commented by john santu last updated on 03/Apr/20
lim_(x→0)  ((e^x −e^(−x) )/(2x)) = lim_(x→0)  ((e^x +e^(−x) )/2) = 1
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} −\mathrm{e}^{−\mathrm{x}} }{\mathrm{2x}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} }{\mathrm{2}}\:=\:\mathrm{1} \\ $$
Commented by mathmax by abdo last updated on 03/Apr/20
we have e^x  =1+(x/(1!)) +(x^2 /(2!)) +o(x^3 )  (x→0)  e^(−x) =1−(x/(1!)) +(x^2 /(2!)) +o(x^3 ) ⇒e^x  +e^(−x) −2 = x^2  +o(x^3 ) ⇒  ((e^x  +e^(−x) −1)/x^2 ) =1+o(x) ⇒lim_(x→0)    ((e^x  +e^(−x) −2)/x^2 )=1
$${we}\:{have}\:{e}^{{x}} \:=\mathrm{1}+\frac{{x}}{\mathrm{1}!}\:+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\:+{o}\left({x}^{\mathrm{3}} \right)\:\:\left({x}\rightarrow\mathrm{0}\right) \\ $$$${e}^{−{x}} =\mathrm{1}−\frac{{x}}{\mathrm{1}!}\:+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\:+{o}\left({x}^{\mathrm{3}} \right)\:\Rightarrow{e}^{{x}} \:+{e}^{−{x}} −\mathrm{2}\:=\:{x}^{\mathrm{2}} \:+{o}\left({x}^{\mathrm{3}} \right)\:\Rightarrow \\ $$$$\frac{{e}^{{x}} \:+{e}^{−{x}} −\mathrm{1}}{{x}^{\mathrm{2}} }\:=\mathrm{1}+{o}\left({x}\right)\:\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{e}^{{x}} \:+{e}^{−{x}} −\mathrm{2}}{{x}^{\mathrm{2}} }=\mathrm{1} \\ $$
Answered by $@ty@m123 last updated on 03/Apr/20
lim_(x→0)  ((e^x +(1/e^x )−2)/x^2 )  lim_(x→0)  ((e^(2x) +1−2e^x )/(x^2 .e^x ))  lim_(x→0)  (((e^x −1)^2 )/(x^2 .e^x ))  lim_(x→0)  (((e^x −1)/x))^2 ×(1/(lim_(x→0)   e^x ))  =1^2 ×1  =1
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\frac{\mathrm{1}}{{e}^{{x}} }−\mathrm{2}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{2}{x}} +\mathrm{1}−\mathrm{2}{e}^{{x}} }{\mathrm{x}^{\mathrm{2}} .{e}^{{x}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left({e}^{{x}} −\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} .{e}^{{x}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{{e}^{{x}} −\mathrm{1}}{{x}}\right)^{\mathrm{2}} ×\frac{\mathrm{1}}{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:{e}^{{x}} } \\ $$$$=\mathrm{1}^{\mathrm{2}} ×\mathrm{1} \\ $$$$=\mathrm{1} \\ $$

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