Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(\sqrt{1+2x}-\sqrt{1-2x})}{\sin x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(a+x{)}^2\sin (a+x)-a^2\sin a}{x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{\left(e^{3+x}-\sin x-e^3\right)}{x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{\left(e^{\tan x}-1\right)}{\tan x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{\left(e^{\tan x}-1\right)}{x}$.