Question

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

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

2021-03-07

Given:
$$\lim_{(x,y)\rightarrow(2,0)}\frac{1-\cos y}{xy^2}$$
we will choose $$y=mx$$ when $$y\rightarrow0\ then\ x\rightarrow0.$$.
$$\lim_{(x,y)\rightarrow(2,0)}\frac{1-\cos y}{xy^2}$$
$$=\lim_{x\rightarrow0}\frac{1-\cos(mx)}{x(mx)^2}$$
$$=\lim_{x\rightarrow0}\frac{m\sin mx}{3m^2x^2}$$
$$=\lim_{x\rightarrow0}\frac{m\cos mx}{6mx}$$
$$=\lim_{x\rightarrow0}\frac{-m\sin mx}{6}$$
$$=0$$