Which of the following is true

If  $\left|ƒ\left(x\right)\right|$ is continuous at $x=a$ then at $x=a$ limit need not exist for $ƒ\left(x\right)$

For every continuous functions $ƒ\left(x\right),g\left(x\right)$ on $(0,\infty )$ satisfying $ƒ\left(x\right)<g\left(x\right)$ for all $x>0$ and both $\underset{x\to \infty }{\lim }ƒ\left(x\right)and\underset{x\to \infty }{\lim }g\left(x\right)$ exist finitely

If both functions $ƒ\left(x\right)andg\left(x\right)$ have limits at $x=a,$ then $\underset{x\to a}{Lt}\left[ƒ\left(g\left(x\right)\right)\right]=\left[\underset{x\to a}{Lt}ƒ\left(g\left(x\right)\right)\right] [ ]$ denotes G.I.F

for every continuous monotonic functions $f\left(x\right)$ , $g\left(x\right)$  on $ℝ$  , Then their sum $ƒ\left(x\right)+g\left(x\right)$  is monotonic on  $ℝ$