What is the limit as (x,y) approaches (0,0) of (sin(xy))/(sqrt(x^2+y^2)) ?

The limit as (x,y) approaches (0,0) of (sin(xy))/(sqrt(x^2+y^2)) is 0

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limit as (x,y) approaches (0,0) of (sin(xy))/(sqrt(x^2+y^2))

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

0

Solution steps

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

Convert to polar coordinates

x = r cos (θ)

= r \to 0 lim \frac{sin ( r cos ( θ ) r sin ( θ ) )}{( r cos ( θ ) ) ^{2} + ( r sin ( θ ) ) ^{2}}

x \to a - lim f (x) = x \to a + lim f (x) = L

then

x \to a lim f (x) = L

r \to 0 - lim \frac{sin ( r cos ( θ ) r sin ( θ ) )}{( r cos ( θ ) ) ^{2} + ( r sin ( θ ) ) ^{2}} = 0

r \to 0 + lim \frac{sin ( r cos ( θ ) r sin ( θ ) )}{( r cos ( θ ) ) ^{2} + ( r sin ( θ ) ) ^{2}} = 0

= 0

\int \frac{x}{x ^{4} + 12 x ^{3} + 36 x ^{2}} d x

\int_{- 2}^{2} (32 - 8 y^{2}) d y

\frac{d}{d x} ((\frac{3 x - 4}{5 x + 2})^{- 5})

f (x) = 5 x^{2} cos (x) + 4 x

\int_{1}^{- 1} 3 e^{x} d x

What is the limit as (x,y) approaches (0,0) of (sin(xy))/(sqrt(x^2+y^2)) ?
The limit as (x,y) approaches (0,0) of (sin(xy))/(sqrt(x^2+y^2)) is 0