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lim-x-0-1-x-1-2x-2020-




Question Number 85293 by jagoll last updated on 20/Mar/20
lim_(x→0^+  )  (√(1/x))−(√((1/(2x))−2020)) ?
$$\underset{{x}\rightarrow\mathrm{0}^{+} \:} {\mathrm{lim}}\:\sqrt{\frac{\mathrm{1}}{\mathrm{x}}}−\sqrt{\frac{\mathrm{1}}{\mathrm{2x}}−\mathrm{2020}}\:? \\ $$
Commented by jagoll last updated on 20/Mar/20
what is result? 0 or ∞?
$$\mathrm{what}\:\mathrm{is}\:\mathrm{result}?\:\mathrm{0}\:\mathrm{or}\:\infty? \\ $$
Answered by mr W last updated on 20/Mar/20
=(((1/(2x))+2020)/( (√(1/x))+(√((1/(2x))−2020))))  =((((1/2)+2020x)/(1+(√((1/2)−2020x)))))(1/( (√x)))  =(1/(1+(1/( (√2)))))×(1/0)  =+∞
$$=\frac{\frac{\mathrm{1}}{\mathrm{2x}}+\mathrm{2020}}{\:\sqrt{\frac{\mathrm{1}}{\mathrm{x}}}+\sqrt{\frac{\mathrm{1}}{\mathrm{2x}}−\mathrm{2020}}} \\ $$$$=\left(\frac{\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{2020}{x}}{\mathrm{1}+\sqrt{\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2020}{x}}}\right)\frac{\mathrm{1}}{\:\sqrt{{x}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}×\frac{\mathrm{1}}{\mathrm{0}} \\ $$$$=+\infty \\ $$
Commented by jagoll last updated on 20/Mar/20
different way, but same the result
$$\mathrm{different}\:\mathrm{way},\:\mathrm{but}\:\mathrm{same}\:\mathrm{the}\:\mathrm{result} \\ $$
Answered by jagoll last updated on 20/Mar/20
Commented by prakash jain last updated on 20/Mar/20
Step 4:  You have used   lim f(x)×g(x)=limf(x)×lim g(x)  this isn′t always valid.
$$\mathrm{Step}\:\mathrm{4}: \\ $$$$\mathrm{You}\:\mathrm{have}\:\mathrm{used}\: \\ $$$$\mathrm{lim}\:{f}\left({x}\right)×{g}\left({x}\right)={limf}\left({x}\right)×{lim}\:{g}\left({x}\right) \\ $$$$\mathrm{this}\:\mathrm{isn}'\mathrm{t}\:\mathrm{always}\:\mathrm{valid}. \\ $$

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