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Author: Tinku Tara

for-any-natural-numbers-m-n-then-m-n-or-m-lt-n-or-m-gt-n-prove-

Question Number 215939 by mokys last updated on 21/Jan/25 $${for}\:{any}\:{natural}\:{numbers}\:{m},{n}\:{then}\:{m}={n}\:{or}\:{m}<{n}\:{or}\:{m}>{n}\:?\:{prove} \\ $$ Answered by MrGaster last updated on 22/Jan/25 $$\forall{m},{n}\in\mathbb{N} \\ $$$${m}={n}\vee{m}<{n}\vee{m}>{n} \\ $$$$\because\mathbb{N}\subset\mathbb{R}\wedge\mathbb{R}\:\mathrm{is}\:\mathrm{totally}\:\mathrm{ordered} \\…

Find-1-lim-x-0-1-cos2x-x-2-2-n-1-n-n-2-3-n-

Question Number 215893 by hardmath last updated on 20/Jan/25 $$\mathrm{Find}: \\ $$$$\left.\mathrm{1}\right)\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}\:−\:\mathrm{cos2x}}{\mathrm{x}^{\mathrm{2}} }\:=\:? \\ $$$$\left.\mathrm{2}\right)\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}=\infty} {\sum}}\:\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{3}^{\boldsymbol{\mathrm{n}}} }\:=\:? \\ $$ Answered by mathmax…

Find-lim-h-0-x-h-3-x-3-h-

Question Number 215889 by hardmath last updated on 20/Jan/25 $$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{h}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}\:+\:\mathrm{h}\right)^{\mathrm{3}} \:+\:\mathrm{x}^{\mathrm{3}} }{\mathrm{h}}\:=\:? \\ $$ Commented by mr W last updated on 20/Jan/25 $$\rightarrow\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{0}}\rightarrow\infty…

lim-x-0-cos-2x-cos-6x-1-cos-3x-cos-5x-

Question Number 215884 by golsendro last updated on 20/Jan/25 $$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:\mathrm{6x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{5x}}\:=? \\ $$$$\:\:\:\: \\ $$ Answered by oubiji last updated on 20/Jan/25 $$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:\mathrm{6x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{5x}}\: \\…

Given-m-gt-0-n-gt-0-m-n-a-Find-the-range-of-a-such-that-m-1-m-n-1-n-gets-its-minimum-iff-m-n-

Question Number 215883 by CrispyXYZ last updated on 20/Jan/25 $$\mathrm{Given}\:{m}>\mathrm{0},\:{n}>\mathrm{0},\:{m}+{n}=\sqrt{{a}}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$“\left({m}+\frac{\mathrm{1}}{{m}}\right)\left({n}+\frac{\mathrm{1}}{{n}}\right)\:\mathrm{gets}\:\mathrm{its}\:\mathrm{minimum}\:\mathrm{iff}\:{m}={n}''. \\ $$ Commented by MathematicalUser2357 last updated on 21/Jan/25 $$“{iff}''?\:{does}\:{it}\:{mean}\:{ifh}.{cc}? \\…