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Category: Limits

advanced-calculus-evaluation-k-2-1-k-k-1-k-k-2-1-k-k-1-k-k-2-1-k-k-n-2-1-n-k-

Question Number 133068 by mnjuly1970 last updated on 18/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{advanced}…..{calculus}…. \\ $$$$\:\:\:{evaluation}::\:\underset{{k}=\mathrm{2}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{k}} \left(\:\frac{\zeta\left({k}\right)−\mathrm{1}}{{k}}\right) \\ $$$$\:\:\:\::::\boldsymbol{\Phi}=\underset{{k}=\mathrm{2}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{k}} \:\frac{\zeta\left({k}\right)−\mathrm{1}}{{k}} \\ $$$$\:\:\:\:\:\:\:\:\:=\underset{{k}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:\underset{{n}=\mathrm{2}}…

Lim-x-pi-2-sin-x-cos-x-x-

Question Number 132971 by metamorfose last updated on 17/Feb/21 $$\:\:{Lim}_{{x}\rightarrow\left(\frac{\pi}{\mathrm{2}}\right)^{−\:\:\:} } \frac{\lfloor{sin}\left({x}\right)\rfloor}{{cos}\left({x}\lfloor{x}\rfloor\right)} \\ $$ Answered by mnjuly1970 last updated on 18/Feb/21 $${ans}:\frac{\mathrm{0}}{\mathrm{0}^{+} }=\mathrm{0} \\ $$…

k-1-1-k-ln-1-1-k-

Question Number 132928 by metamorfose last updated on 17/Feb/21 $$\underset{{k}=\mathrm{1}} {\overset{+\infty} {\sum}}\left(−\mathrm{1}\right)^{{k}} {ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{k}}\right) \\ $$ Commented by Olaf last updated on 17/Feb/21 $${sorry}\:{sir},\:{I}\:{deleted}\:{my}\:{answer}. \\ $$$${it}\:{was}\:{wrong}.…

Find-the-limit-of-this-sequence-2-2-2-2-2-2-2-2-2-2-Show-that-the-sum-of-the-terms-of-this-infinite-sequence-does-not-converge-

Question Number 1608 by 112358 last updated on 26/Aug/15 $${Find}\:{the}\:{limit}\:{of}\:{this}\:{sequence}. \\ $$$$\sqrt{\mathrm{2}},\sqrt{\mathrm{2}\sqrt{\mathrm{2}}},\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}}}},\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}}}}},… \\ $$$${Show}\:{that}\:{the}\:{sum}\:{of}\:{the} \\ $$$${terms}\:{of}\:{this}\:{infinite}\:{sequence} \\ $$$${does}\:{not}\:{converge}. \\ $$ Commented by Rasheed Ahmad last…

Use-the-definition-of-the-limit-to-show-that-lim-x-3-x-6-x-1-I-m-hoping-to-better-understand-this-concept-by-example-so-please-help-me-by-explaning-the-reasoni

Question Number 1499 by 112358 last updated on 14/Aug/15 $${Use}\:{the}\:\epsilon−\delta\:{definition}\:{of}\:{the} \\ $$$${limit}\:{to}\:{show}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{{x}}{\mathrm{6}−{x}}=\mathrm{1}\:. \\ $$$$\left({I}'{m}\:{hoping}\:{to}\:{better}\:{understand}\right. \\ $$$${this}\:{concept}\:{by}\:{example}\:{so}\:{please} \\ $$$${help}\:{me}\:{by}\:{explaning}\:{the}\: \\ $$$$\left.{reasoning}\:{behind}\:{your}\:{steps}.\right) \\ $$…

If-a-1-then-lim-x-a-2-3x-5-4x-2-3x-1-x-2-8x-2-a-1-b-4-5-c-3-5-d-2-5-e-1-5-

Question Number 132550 by liberty last updated on 15/Feb/21 $$\mathrm{If}\:{a}=\mathrm{1}\:\mathrm{then}\:\underset{{x}\rightarrow\left({a}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{5}\right)} {\mathrm{lim}}\frac{−\mathrm{4}{x}^{\mathrm{2}} +\sqrt{\mathrm{3}{x}+\mathrm{1}}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{8}{x}}+\mathrm{2}}\:=? \\ $$$$\left(\mathrm{a}\right)\:−\mathrm{1}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:−\frac{\mathrm{4}}{\mathrm{5}}\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:−\frac{\mathrm{3}}{\mathrm{5}} \\ $$$$\left(\mathrm{d}\right)\:−\frac{\mathrm{2}}{\mathrm{5}}\:\:\:\:\left(\mathrm{e}\right)\:−\frac{\mathrm{1}}{\mathrm{5}} \\ $$ Commented by MJS_new last updated…