Mathematics - Limits Question with Solution | TestHub

S₁: If ${\lim }_{x\to a}f(x)+g(x)$ and ${\lim }_{x\to a}f(x)-g(x)$ exist, then it is not necessary that ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist separately.

S₂: If ${\lim }_{x\to a}(f(x)+g(x))$ exists, then it is necessary that ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist separately.

S₃: ${\lim }_{x\to a}\left(\frac{f(x)}{g(x)}\right)=e^{{\lim }_{x\to a}(f(x)-1)g(x)}$.

S₄: ${\lim }_{x\to 0^{+}}\left[\frac{e^{\cos x}-e^{\ln x}}{1-\ln x}\right]=1$, where $[.]$ represents the greatest integer function.

State in order whether S₁, S₂, S₃, S₄ are true or false.