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Question Number 379 by userid1 last updated on 25/Jan/15

Findlim_(x→0) (((e^(1/x) −1)/(e^(1/x) +1)))

$$\mathrm{Find}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{e}^{\mathrm{1}/{x}} −\mathrm{1}}{{e}^{\mathrm{1}/{x}} +\mathrm{1}}\right) \\ $$

Commented by 123456 last updated on 25/Dec/14

lim_(x→0+) ((e^(1/x) −1)/(e^(1/x) +1))=^? lim_(x→0+) ((−(1/x^2 )e^(1/x) )/(−(1/x^2 )e^(1/x) ))=^? 1(→(∞/∞)) lim_(x→0−) ((e^(1/x) −1)/(e^(1/x) +1))=^? ((0−1)/(0+1))=^? −1 lim_(x→0) ((e^(1/x) −1)/(e^(1/x) +1))=^? ∄

$$\underset{{x}\rightarrow\mathrm{0}+} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{e}^{\frac{\mathrm{1}}{{x}}} +\mathrm{1}}\overset{?} {=}\underset{{x}\rightarrow\mathrm{0}+} {\mathrm{lim}}\frac{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{e}^{\frac{\mathrm{1}}{{x}}} }{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{e}^{\frac{\mathrm{1}}{{x}}} }\overset{?} {=}\mathrm{1}\left(\rightarrow\frac{\infty}{\infty}\right) \\ $$$$\underset{{x}\rightarrow\mathrm{0}−} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{e}^{\frac{\mathrm{1}}{{x}}} +\mathrm{1}}\overset{?} {=}\frac{\mathrm{0}−\mathrm{1}}{\mathrm{0}+\mathrm{1}}\overset{?} {=}−\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{e}^{\frac{\mathrm{1}}{{x}}} +\mathrm{1}}\overset{?} {=}\nexists \\ $$

Answered by prakash jain last updated on 27/Dec/14

As given in comments for given function lim_(x→0^+ ) f(x)≠lim_(x→0^− ) f(x) so the limit is not defined

$$\mathrm{A}{s}\:\mathrm{given}\:\mathrm{in}\:\mathrm{comments}\:\mathrm{for}\:\mathrm{given}\:\mathrm{function} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:{f}\left({x}\right)\neq\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:{f}\left({x}\right) \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\ $$

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