Question Number 116939 by joki last updated on 08/Oct/20
$${prof}\:\:{if}\:\:{the}\:{limitf}\left({x}\right)={L}\:{and}\: \\ $$$${limit}\:{f}\left({x}\right)={M},{then}\:{L}={M} \\ $$
Answered by MAB last updated on 08/Oct/20
$$\underset{\boldsymbol{\mathrm{x}}\rightarrow{x}_{\mathrm{0}} } {\boldsymbol{\mathrm{lim}}}{f}\left({x}\right)={L}={M} \\ $$$$\Rightarrow\left[\left(\forall\varepsilon>\mathrm{0}\right)\left(\exists\alpha>\mathrm{0}\right)\left(\forall{x}\in{D}_{{f}} \right):\mid{x}−{x}_{\mathrm{0}} \mid<\alpha\Rightarrow\mid{f}\left({x}\right)−{L}\:\mid<\varepsilon\right] \\ $$$${and}\:\left[\left(\forall\varepsilon'>\mathrm{0}\right)\left(\exists\alpha'>\mathrm{0}\right)\left(\forall{x}\in{D}_{{f}} \right):\mid{x}−{x}_{\mathrm{0}} \mid<\alpha'\Rightarrow\mid{f}\left({x}\right)−{M}\:\mid<\varepsilon'\right] \\ $$$${we}\:{have}\:\mid{M}−{L}\mid\leqslant\mid{f}\left({x}\right)−{L}\mid+{f}\left({x}\right)−{M}\mid\:{triangle}\:{inequality} \\ $$$${hence}\:\left(\forall\epsilon>\mathrm{0}\right)\left(\exists\delta={min}\left(\alpha,\alpha'\right)>\mathrm{0}\right):\:\mid{x}−{x}_{\mathrm{0}} \mid<\delta\Rightarrow\mid{M}−{L}\mid<\backepsilon \\ $$$${finally}\:{M}={L} \\ $$
Answered by MAB last updated on 08/Oct/20
$$\underset{\boldsymbol{\mathrm{x}}\rightarrow{x}_{\mathrm{0}} } {\boldsymbol{\mathrm{lim}}}{f}\left({x}\right)={L}={M} \\ $$$$\Rightarrow\left[\left(\forall\varepsilon>\mathrm{0}\right)\left(\exists\alpha>\mathrm{0}\right)\left(\forall{x}\in{D}_{{f}} \right):\mid{x}−{x}_{\mathrm{0}} \mid<\alpha\Rightarrow\mid{f}\left({x}\right)−{L}\:\mid<\varepsilon\right] \\ $$$${and}\:\left[\left(\forall\varepsilon'>\mathrm{0}\right)\left(\exists\alpha'>\mathrm{0}\right)\left(\forall{x}\in{D}_{{f}} \right):\mid{x}−{x}_{\mathrm{0}} \mid<\alpha'\Rightarrow\mid{f}\left({x}\right)−{M}\:\mid<\varepsilon'\right] \\ $$$${we}\:{have}\:\mid{M}−{L}\mid\leqslant\mid{f}\left({x}\right)−{L}\mid+{f}\left({x}\right)−{M}\mid\:{triangle}\:{inequality} \\ $$$${hence}\:\left(\forall\epsilon>\mathrm{0}\right)\left(\exists\delta={min}\left(\alpha,\alpha'\right)>\mathrm{0}\right):\:\mid{x}−{x}_{\mathrm{0}} \mid<\delta\Rightarrow\mid{M}−{L}\mid<\backepsilon \\ $$$${finally}\:{M}={L} \\ $$