Given:

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}\)

On plugging in the value (x,y)=(0,0)

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}=\cos(\frac{0^2+0^3}{0+0+1})\)

On simplifying

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}=1\)

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}\)

On plugging in the value (x,y)=(0,0)

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}=\cos(\frac{0^2+0^3}{0+0+1})\)

On simplifying

\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}=1\)