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show-that-if-f-is-a-differentiable-function-at-the-point-x-a-then-f-is-continuous-at-x-a-

Question Number 71769 by psyche last updated on 19/Oct/19 $${show}\:{that}\:{if}\:{f}\:{is}\:{a}\:{differentiable}\:{function}\:{at}\:{the}\:{point}\:{x}={a},\:{then}\:{f}\:{is}\:{continuous}\:{at}\:{x}={a}. \\ $$ Commented by kaivan.ahmadi last updated on 19/Oct/19 $${if}\:{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)\neq{f}\left({a}\right)\:\Rightarrow{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)−{f}\left({a}\right)\neq\mathrm{0}\Rightarrow \\ $$$${then}\:{f}'\left({a}\right)={lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}=+\infty\vee−\infty…

from-the-figure-above-the-square-S-s-diameter-length-is-increasing-by-25-m-s-to-the-north-east-initially-at-length-30-2-m-and-circle-C-s-radius-is-decreasing-by-2-m-s-initially-at-length-100-m-kn

Question Number 137303 by Raxreedoroid last updated on 31/Mar/21 $$\mathrm{from}\:\mathrm{the}\:\mathrm{figure}\:\mathrm{above} \\ $$$${the}\:{square}\:{S}'\mathrm{s}\:\mathrm{diameter}\:\mathrm{length}\:\mathrm{is}\:\mathrm{increasing} \\ $$$$\mathrm{by}\:\mathrm{25}\:\mathrm{m}/\mathrm{s}\:\mathrm{to}\:\mathrm{the}\:\mathrm{north}−\mathrm{east}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{length}\:\mathrm{30}\sqrt{\mathrm{2}\:}\mathrm{m}\:\mathrm{and}\:\mathrm{circle}\: \\ $$$${C}'\mathrm{s}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{decreasing}\:\mathrm{by}\:\mathrm{2}\:\mathrm{m}/\mathrm{s}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{length}\:\mathrm{100}\:\mathrm{m} \\ $$$$\mathrm{knowing}\:\mathrm{that}\:\mathrm{the}\:\mathrm{blue}\:\mathrm{line}'\mathrm{s}\:\mathrm{length}\:=\:\mathrm{40m} \\ $$$$\mathrm{at}\:\mathrm{what}\:\mathrm{time}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{point}\:{p} \\ $$$$\mathrm{and}\:\mathrm{point}\:{q}\:\mathrm{will}\:\mathrm{equal}\:\mathrm{0}? \\ $$$$\mathrm{and}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{distance}\:\mathrm{at}\:\mathrm{that}\:\mathrm{time}? \\…

Find-at-least-the-first-four-non-zero-term-in-a-power-series-expansion-about-x-0-for-a-general-solution-to-z-x-2-z-0-

Question Number 71761 by TawaTawa last updated on 19/Oct/19 $$\mathrm{Find}\:\mathrm{at}\:\mathrm{least}\:\mathrm{the}\:\mathrm{first}\:\mathrm{four}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{term}\:\mathrm{in}\:\mathrm{a}\:\mathrm{power} \\ $$$$\mathrm{series}\:\mathrm{expansion}\:\mathrm{about}\:\:\mathrm{x}\:\:=\:\:\mathrm{0}\:\:\mathrm{for}\:\mathrm{a}\:\mathrm{general}\:\mathrm{solution} \\ $$$$\mathrm{to}\:\:\:\:\mathrm{z}''\:\:−\:\:\mathrm{x}^{\mathrm{2}} \mathrm{z}\:\:\:=\:\:\mathrm{0} \\ $$ Commented by mathmax by abdo last updated on…

Question-71759

Question Number 71759 by naka3546 last updated on 19/Oct/19 Commented by Prithwish sen last updated on 19/Oct/19 $$\boldsymbol{\mathrm{x}}=\:\frac{\mathrm{1}}{\mathrm{3}}.\frac{\mathrm{1}}{\mathrm{3}}.\frac{\mathrm{2}}{\mathrm{4}}.\frac{\mathrm{3}}{\mathrm{5}}………\frac{\mathrm{998}}{\mathrm{1000}}.\frac{\mathrm{999}}{\mathrm{1001}}\:=\:\frac{\mathrm{2}×\mathrm{1001001}}{\mathrm{3×1000}×\mathrm{1001}}\approx\mathrm{0}.\mathrm{67} \\ $$$$\therefore\:\mathrm{100x}\:\approx\:\mathrm{67} \\ $$$$\boldsymbol{\mathrm{By}}\:\boldsymbol{\mathrm{considering}} \\ $$$$\frac{\boldsymbol{\mathrm{n}}^{\mathrm{3}} −\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{3}}…