Almightytay1901 Almightytay1901
  • 30-10-2017
  • Mathematics
contestada

Find the given limit lim(x,y)→(a,a)x4−y4x2−y2=lim(x,y)→(a,a) (x^4−y^4)/(x2−y2)= here aa is a constant. (find the limit assuming x≠y)

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LammettHash
LammettHash LammettHash
  • 30-10-2017
If [tex]x\neq y[/tex], then we can write

[tex]\dfrac{x^4-y^4}{x^2-y^2}=\dfrac{(x^2-y^2)(x^2+y^2)}{x^2-y^2}=x^2+y^2[/tex]

and since this is continuous for all [tex]x\neq y[/tex], the limit as [tex](x,y)\to(a,a)[/tex] will be

[tex]\displaystyle\lim_{(x,y)\to(a,a)}\frac{x^4-y^4}{x^2-y^2}=\lim_{(x,y)\to(a,a)}(x^2+y^2)=2a^2[/tex]
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