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Category: Limits

BeMath-1-lim-x-0-1-cos-x-cos-2x-cos-3x-cos-nx-x-2-2-x-2-y-xy-4y-0-y-1-2-and-y-1-0-3-find-the-probability-that-a-person-throwing-three-coins-at-once-

Question Number 108469 by bemath last updated on 17/Aug/20 $$\:\:\:\frac{\subset\mathcal{B}{e}\mathcal{M}{ath}\supset}{\cap} \\ $$$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:\sqrt{\mathrm{cos}\:\mathrm{3}{x}}…\sqrt{\mathrm{cos}\:{nx}}}{{x}^{\mathrm{2}} }\:? \\ $$$$\left(\mathrm{2}\right){x}^{\mathrm{2}} {y}''+{xy}'−\mathrm{4}{y}=\mathrm{0};\:{y}\left(\mathrm{1}\right)=\mathrm{2}\:{and} \\ $$$$\:\:\:{y}'\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\left(\mathrm{3}\right){find}\:{the}\:{probability}\:{that}\:{a}\:{person}\:\:{throwing}\:{three} \\ $$$${coins}\:{at}\:{once}\:{will}\:{get}\:{all}\:{the}\:{face}\:{or}\: \\ $$$${everything}\:{back}\:{for}\:{second}\:{time}\:{at}…

Suppose-that-f-and-g-are-two-functions-such-that-lim-x-a-g-x-0-and-lim-x-a-f-x-g-x-exist-Prove-that-lim-x-a-f-x-0-

Question Number 42933 by Joel578 last updated on 05/Sep/18 $$\mathrm{Suppose}\:\mathrm{that}\:{f}\:\mathrm{and}\:{g}\:\mathrm{are}\:\mathrm{two}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{g}\left({x}\right)\:=\:\mathrm{0}\:\:\:\:\mathrm{and}\:\:\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}\:\:\:\mathrm{exist}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$ Commented by MrW3 last updated on…

Question-42897

Question Number 42897 by Joel578 last updated on 04/Sep/18 Commented by Joel578 last updated on 04/Sep/18 $$\mathrm{For}\:\mathrm{question}\:\left({c}\right), \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{G}\left({x}\right)\:=\:\mathrm{2}\:\:\:\mathrm{or}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{G}\left({x}\right)\:=\:\mathrm{0}\:? \\ $$ Answered by…

Question-42895

Question Number 42895 by Joel578 last updated on 04/Sep/18 Commented by Joel578 last updated on 04/Sep/18 $$\mathrm{For}\:\mathrm{question}\:\left({c}\right), \\ $$$$\mathrm{the}\:\mathrm{limit}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}\:\mathrm{or}\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{1}\:? \\ $$ Answered by MJS…

Derive-Leibniz-s-formula-fg-n-x-0-k-0-n-C-n-k-f-k-x-0-g-n-k-x-0-

Question Number 108349 by Ar Brandon last updated on 16/Aug/20 $$\mathrm{Derive}\:\mathrm{Leibniz}'\mathrm{s}\:\mathrm{formula}\:: \\ $$$$\left(\mathrm{fg}\right)^{\left(\mathrm{n}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \mathrm{f}^{\left(\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)\mathrm{g}^{\left(\mathrm{n}−\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right) \\ $$…