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Category: Limits

lim-x-0-1-cos-x-cos-2x-x-2-

Question Number 144556 by imjagoll last updated on 26/Jun/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}^{\mathrm{2}} }\right)\:=? \\ $$ Answered by liberty last updated on 26/Jun/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}^{\mathrm{2}} }\right)=? \\…

lim-x-2x-3x-2-3x-x-1-

Question Number 144558 by imjagoll last updated on 26/Jun/21 $$\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\left(\frac{\mathrm{2x}−\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{3x}}}{\mathrm{x}−\mathrm{1}}\right)=? \\ $$ Answered by liberty last updated on 26/Jun/21 $$\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\left(\frac{\mathrm{2x}−\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{3x}}}{\mathrm{x}−\mathrm{1}}\right)=? \\…

Question-144513

Question Number 144513 by mnjuly1970 last updated on 26/Jun/21 Answered by Olaf_Thorendsen last updated on 26/Jun/21 $$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{−{x}} \:=\:{e}^{−{x}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)} \:\underset{\infty} {\sim}\:{e}^{−{x}\left(\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{−{x}} \:\underset{\infty} {\sim}\:\frac{\mathrm{1}}{{e}}{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}}…

it-is-known-that-after-injecting-the-1-st-dosage-of-a-drug-in-exactly-hours-p-t-Ae-t-24-0-t-8-A-Ae-1-3-e-t-8-24-8-t-24-A-is-the-initial-dosage-at-t-0h

Question Number 144502 by alcohol last updated on 26/Jun/21 $${it}\:{is}\:{known}\:{that}\:{after}\:{injecting}\:{the}\:\mathrm{1}^{{st}} \\ $$$${dosage}\:{of}\:{a}\:{drug}\:{in}\:{exactly}\:{hours} \\ $$$${p}\left({t}\right)=\begin{cases}{{Ae}^{−\frac{{t}}{\mathrm{24}}} ,\:\mathrm{0}\leqslant{t}\leqslant\mathrm{8}}\\{\left({A}+{Ae}^{−\frac{\mathrm{1}}{\mathrm{3}}} \right){e}^{−\left(\frac{{t}−\mathrm{8}}{\mathrm{24}}\right)} ,\:\mathrm{8}\:\leqslant\:{t}\leqslant\mathrm{24}}\end{cases} \\ $$$${A}\:{is}\:{the}\:{initial}\:{dosage}\:{at}\:{t}=\:\mathrm{0}{hours} \\ $$$$\left.{a}\right)\:{show}\:{that}\:{towards}\:{the}\:{application}\:{of} \\ $$$${the}\:{n}^{{th}} \:{dosage}\:,\:{the}\:{dosage}\:{remained}\:{in}\:{blood}\:{is} \\…

lim-x-1-1-4-4-x-1-5-5-x-1-5-

Question Number 144413 by imjagoll last updated on 25/Jun/21 $$\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left[\:\frac{\mathrm{1}}{\mathrm{4}−\mathrm{4}\sqrt{\mathrm{x}}}−\frac{\mathrm{1}}{\mathrm{5}−\mathrm{5}\sqrt[{\mathrm{5}}]{\mathrm{x}}}\:\right]\:=? \\ $$ Answered by Olaf_Thorendsen last updated on 25/Jun/21 $${f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}−\mathrm{4}\sqrt{{x}}}−\frac{\mathrm{1}}{\mathrm{5}−\mathrm{5}\sqrt[{\mathrm{5}}]{{x}}} \\ $$$$\mathrm{Let}\:{x}\:=\:{X}^{\mathrm{10}} \\ $$$${f}\left({X}^{\mathrm{10}}…