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Find the following limits. Be sure to explain your reasoning at every step.
\begin{alignat*}{3}
&a)    \lim_{x \to \infty}\frac{x+\sin x }{ 5x+6} &              &b)    \lim_
{x \to \infty}\frac{\sin x }{ x} &              &c)    \lim_{x \to \infty}x \sin \frac{
1 }{ x} \\
&d)    \lim_{x \to \infty}\frac{x \sin x }{ x^2+5} &             
&e) \lim_{x \to \infty}{\sqrt {x^2+x} - x} &             
&f) \lim_{x \to \infty}\frac{x^2( 1+ \sin^2x) }{ (x+\sin x
)^2}
\end{alignat*}