Use the method of your choice to evaluate the following limits. lim_{(x,y)rightarrow(2,0)}frac{1-cos y}{xy^2}

Question

Use the method of your choice to evaluate the following limits.

lim_{(x, y) \to (2, 0)} \frac{1 - \cos y}{x y^{2}}

Answer 1

Given:
$lim_{(x, y) \to (2, 0)} \frac{1 - \cos y}{x y^{2}}$
we will choose $y = m x$ when $y \to 0 t h e n x \to 0.$ .
$lim_{(x, y) \to (2, 0)} \frac{1 - \cos y}{x y^{2}}$
$= lim_{x \to 0} \frac{1 - \cos (m x)}{x (m x)^{2}}$
$= lim_{x \to 0} \frac{m \sin m x}{3 m^{2} x^{2}}$
$= lim_{x \to 0} \frac{m \cos m x}{6 m x}$
$= lim_{x \to 0} \frac{- m \sin m x}{6}$
$= 0$