2.4 演習問題～極限，連続

問 2.21 (極限) 次の極限を求めよ．
    (1)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2}{x^2+y^2}}$     (2)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2y}{x^2+y^2}}$     (3)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^3+x^2y}{2x^2+y^2}}$     (4)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}}$
    (5)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2-2y^2}{x^2+y^2}}$     (6)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4}}$     (7)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^3-y^3}{x^2+y^2}}$     (8)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x+y+1}{x^2+y^2+1}}$
    (9)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2+2y^2}{2x^2+y^2}}$     (10)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{2x^3-y^3+x^2+y^2}{x^2+y^2}}$     (11)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{\sqrt{x^2+y^2}}}$
    (12)   $\displaystyle{\lim_{(x,y)\to(0,0)}\frac{x\sqrt{\vert y\vert}}{\sqrt{x^2+y^2}}}$

問 2.22 (連続) 次の関数

が原点で連続となるか議論せよ．ただし

は適当に定義せよ．
    (1)   $\displaystyle{\frac{x^4+x^2+y^2+y^3}{x^2+y^2}}$     (2)   $\displaystyle{\frac{x^3+y^3}{x^2+y^2}}$     (3)   $\displaystyle{\frac{x^2-y^2}{x^2+y^2}}$     (4)   $\displaystyle{\frac{xy}{x^2+y^2}}$     (5)   $\displaystyle{\frac{x+y}{x^2+y^2}}$     (6)   $\displaystyle{\frac{x^2y^2}{x^4+y^4}}$
    (7)   $\displaystyle{\frac{x^2+y^2}{x^2+2y^2}}$     (8)   $\displaystyle{\frac{x^4-3x^2y^2}{2x^2+y^2}}$     (9)   $\displaystyle{\frac{e^{x^2+y^2}-1}{x^2+y^2}}$     (10)   $\displaystyle{xy\log(x^2+y^2)}$     (11)   $\displaystyle{\frac{x}{\sqrt{x^2+y^2}}}$
    (12)   $\displaystyle{\frac{\sin(x+y)}{x+y}}$     (13)   $\displaystyle{(x+y)\cos\frac{y}{x}}$

2.4 演習問題 ～ 極限，連続

2.4 演習問題～極限，連続