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Find the following limits. Be sure to explain your reasoning at each 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*}