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Question Number 12062 by tawa last updated on 10/Apr/17

lim_(x→y)   ((x^n  − y^n )/(x − y))

$$\underset{{x}\rightarrow\mathrm{y}} {\mathrm{lim}}\:\:\frac{\mathrm{x}^{\mathrm{n}} \:−\:\mathrm{y}^{\mathrm{n}} }{\mathrm{x}\:−\:\mathrm{y}} \\ $$

Answered by ajfour last updated on 11/Apr/17

x^n −y^n =(x−y)(x^(n−1) +yx^(n−2) +               y^2 x^(n−3) +...+y^(n−2) x+y^(n−1) )  lim_(x→y)  ((x^n −y^n )/(x−y)) = y^(n−1) +y.y^(n−2) +               y^2 .y^(n−3) +...+y^(n−2) .y+y^(n−1)             = n(y^(n−1) ) .

$$\boldsymbol{{x}}^{\boldsymbol{{n}}} −\boldsymbol{{y}}^{\boldsymbol{{n}}} =\left(\boldsymbol{{x}}−\boldsymbol{{y}}\right)\left(\boldsymbol{{x}}^{\boldsymbol{{n}}−\mathrm{1}} +\boldsymbol{{yx}}^{\boldsymbol{{n}}−\mathrm{2}} +\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{x}}^{\boldsymbol{{n}}−\mathrm{3}} +...+\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{2}} \boldsymbol{{x}}+\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{1}} \right) \\ $$$$\underset{{x}\rightarrow{y}} {\mathrm{lim}}\:\frac{\boldsymbol{{x}}^{\boldsymbol{{n}}} −\boldsymbol{{y}}^{\boldsymbol{{n}}} }{\boldsymbol{{x}}−\boldsymbol{{y}}}\:=\:\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{1}} +\boldsymbol{{y}}.\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{2}} + \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}^{\mathrm{2}} .\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{3}} +...+\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{2}} .\boldsymbol{{y}}+\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\boldsymbol{{n}}\left(\boldsymbol{{y}}^{\boldsymbol{{n}}−\mathrm{1}} \right)\:. \\ $$

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 11/Apr/17

L^′ Hopital role:  (y=const.)  l=lim_(x→y) ((nx^(n−1) )/1)=ny^(n−1)    .■

$${L}^{'} {Hopital}\:{role}:\:\:\left({y}={const}.\right) \\ $$$${l}=\underset{{x}\rightarrow{y}} {\mathrm{lim}}\frac{{nx}^{{n}−\mathrm{1}} }{\mathrm{1}}={ny}^{{n}−\mathrm{1}} \:\:\:.\blacksquare \\ $$

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