Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 165441 by mnjuly1970 last updated on 01/Feb/22

Answered by mahdipoor last updated on 01/Feb/22

if g(x) is continuous and increment  function in x≥a ⇒lim_(x→a^+ ) ⌊g(x)⌋=⌊g(a)⌋  prove: lim_(x→a^+ ) g(x)=g(a)⇒∀ε>0   ∃σ>0  ,  0<x−a<σ⇒∣g(x)−g(a)∣<ε ⇒  0<g(x)−g(a)<ε⇒  0≤⌊g(x)⌋−⌊g(a)⌋≤g(x)−g(a)<ε ⇒  ∣⌊g(x)⌋−⌊g(a)⌋∣<ε ⇒lim_(x→a^+ ) ⌊g(x)⌋=⌊g(a)⌋  ,  h(x)=nf(x) is continuous and increment  function in x≥c=0.5(3+(√5)) ⇒  lim_(x→c^+ ) ⌊h(x)⌋= ⌊h(c)⌋=⌊(n/2)⌋=3  ⇒n=6 or 7

$$\mathrm{if}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{increment} \\ $$$$\mathrm{function}\:\mathrm{in}\:\mathrm{x}\geqslant\mathrm{a}\:\Rightarrow\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}}\lfloor\mathrm{g}\left(\mathrm{x}\right)\rfloor=\lfloor\mathrm{g}\left(\mathrm{a}\right)\rfloor \\ $$$${prove}:\:\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}g}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{a}\right)\Rightarrow\forall\varepsilon>\mathrm{0}\:\:\:\exists\sigma>\mathrm{0}\:\:, \\ $$$$\mathrm{0}<\mathrm{x}−\mathrm{a}<\sigma\Rightarrow\mid\mathrm{g}\left(\mathrm{x}\right)−\mathrm{g}\left(\mathrm{a}\right)\mid<\varepsilon\:\Rightarrow \\ $$$$\mathrm{0}<\mathrm{g}\left(\mathrm{x}\right)−\mathrm{g}\left(\mathrm{a}\right)<\varepsilon\Rightarrow \\ $$$$\mathrm{0}\leqslant\lfloor\mathrm{g}\left(\mathrm{x}\right)\rfloor−\lfloor\mathrm{g}\left(\mathrm{a}\right)\rfloor\leqslant\mathrm{g}\left(\mathrm{x}\right)−\mathrm{g}\left(\mathrm{a}\right)<\varepsilon\:\Rightarrow \\ $$$$\mid\lfloor\mathrm{g}\left(\mathrm{x}\right)\rfloor−\lfloor\mathrm{g}\left(\mathrm{a}\right)\rfloor\mid<\varepsilon\:\Rightarrow\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}}\lfloor\mathrm{g}\left(\mathrm{x}\right)\rfloor=\lfloor\mathrm{g}\left(\mathrm{a}\right)\rfloor \\ $$$$, \\ $$$${h}\left({x}\right)={nf}\left({x}\right)\:{is}\:{continuous}\:{and}\:{increment} \\ $$$${function}\:{in}\:{x}\geqslant{c}=\mathrm{0}.\mathrm{5}\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)\:\Rightarrow \\ $$$$\underset{{x}\rightarrow{c}^{+} } {\mathrm{lim}}\lfloor{h}\left({x}\right)\rfloor=\:\lfloor{h}\left({c}\right)\rfloor=\lfloor\frac{{n}}{\mathrm{2}}\rfloor=\mathrm{3} \\ $$$$\Rightarrow{n}=\mathrm{6}\:{or}\:\mathrm{7} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com