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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} & \qquad &b)\; \lim_
 {x \to \infty}\frac{\sin x }{ x} & \qquad &c)\; \lim_{x \to \infty}x\,\sin \frac{
 1 }{ x} \\
 &d)\; \lim_{x \to \infty}\frac{x\,\sin x }{ x^2+5} & \qquad
 &e) \lim_{x \to \infty}{\sqrt {x^2+x} - x} & \qquad
 &f) \lim_{x \to \infty}\frac{x^2( 1+ \sin^2x) }{ (x+\sin x
 )^2}
 \end{alignat*}
 
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