PG entrance exam CUET PG 2025 Mock Test Mathematical Science Analysis Functions of Several Variables
For t ∈ R, let [t] denote the greatest integer less than or equal to t. Define functions h: R2 → R and g: R → R by \(h(x, y)=\left\{\begin{array}{ll} \frac{-1}{x^{2}-y} & \text { if } x^{2} \neq y \\ 0 & \text { if } x^{2}=y \end{array} \text { and } g(x)=\left\{\begin{array}{ll} \frac{\sin x}{x} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right.\right.\), Then which one of the following is FALSE?
1
\(\lim _{(x, y) \rightarrow(\sqrt{2}, \pi)} \cos \left(\frac{x^{2} y}{x^{2}+1}\right)=\frac{-1}{2}\)
2
\(\lim _{(x, y) \rightarrow(\sqrt{2}, 2)} e^{h(x, y)}=0\)
3
\(\lim _{(x, y) \rightarrow(e, e)} \ln \left(x^{y-[y]}\right)=e+2 .\)
4
\(\lim _{(x, y) \rightarrow(0,0)} e^{2 y} g(x)=1\)