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lim-x-y-x-n-y-n-x-y-




Question Number 9157 by tawakalitu last updated on 21/Nov/16
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 prakash jain last updated on 21/Nov/16
x^n −y^n =(x−y)(x^(n−1) +x^(n−2) y+..+xy^(n−2) +y^(n−1) )  ((x^n −y^n )/(x−y))=(x−y)(x^(n−1) +x^(n−2) y+..+xy^(n−2) +y^(n−1) )  lim_(x→y) ((x^n −y^n )/(x−y))=(y^(n−1) +y^(n−2) y+..+yy^(n−2) +y^(n−1) )  =ny^(n−1)
$${x}^{{n}} −{y}^{{n}} =\left({x}−{y}\right)\left({x}^{{n}−\mathrm{1}} +{x}^{{n}−\mathrm{2}} {y}+..+{xy}^{{n}−\mathrm{2}} +{y}^{{n}−\mathrm{1}} \right) \\ $$$$\frac{{x}^{{n}} −{y}^{{n}} }{{x}−{y}}=\left({x}−{y}\right)\left({x}^{{n}−\mathrm{1}} +{x}^{{n}−\mathrm{2}} {y}+..+{xy}^{{n}−\mathrm{2}} +{y}^{{n}−\mathrm{1}} \right) \\ $$$$\underset{{x}\rightarrow{y}} {\mathrm{lim}}\frac{{x}^{{n}} −{y}^{{n}} }{{x}−{y}}=\left({y}^{{n}−\mathrm{1}} +{y}^{{n}−\mathrm{2}} {y}+..+{yy}^{{n}−\mathrm{2}} +{y}^{{n}−\mathrm{1}} \right) \\ $$$$={ny}^{{n}−\mathrm{1}} \\ $$
Commented by tawakalitu last updated on 21/Nov/16
wow, thank you sir. God bless you.
$$\mathrm{wow},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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