Find out the answer to "Nonexistence of a limit Investiage the limit lim(x,y)→(0,0)(x+y)2x2+y2"

Jason Farmer

Answered question

2021-01-31

Nonexistence of a limit Investiage the limit

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}}

Answer & Explanation

pivonie8

Skilled2021-02-01Added 91 answers

By applying identity

{(a + b)}^{2} = (a^{2} + b^{2} + 2 a b)

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = lim_{(x, y) \to (0, 0)} \frac{x^{2} + y^{2} + 2 x y}{x^{2} + y^{2}}

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = lim_{(x, y) \to (0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2}} + \frac{2 x y}{x^{2} + y^{2}}

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = lim_{(x, y) \to (0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2}} + lim_{(x, y) \to (0, 0)} \frac{2 x y}{x^{2} + y^{2}}

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = lim_{(x, y) \to (0, 0)} 1 + lim_{(x, y) \to (0, 0)} \frac{2 x y}{x^{2} + y^{2}}

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = 1 + 0

lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}} = 1

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