# "One way to do this is to use the basic (and obvious) fact that:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \lim_{ x rarr infty } 1/x \ = \ 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (I) #
# "We can use this in our example as follows:" #
# \lim_{ x rarr infty } { 2 x^2 + 2 x + 3 } / { x^3 + x^2 + 1 } \ = \ \lim_{ x rarr infty } { 2 x^2 + 2 x + 3 } / { x^3 + x^2 + 1 } cdot {1/x^3} / {1/x^3} #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } { {2 x^2}/{x^3} + {2 x}/{x^3} + {3}/{x^3} } / { {x^3}/{x^3} + {x^2}/{x^3} + {1}/{x^3} } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } { {2}/{x} + {2}/{x^2} + {3}/{x^3} } / { x + {1}/{x} + {1}/{x^3} } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } { 2 ( {1}/{x} ) + 2 ( {1}/{x} )^2 + 3 ( {1}/{x} )^3 } / { x + ( {1}/{x} ) + ( {1}/{x} )^3 } #
# \qquad "now, using eqn. (I) above, and continuing, we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } { 2 ( 0 ) + 2 ( 0 )^2 + 3 ( 0 )^3 } / { x + ( 0 ) + ( 0 )^3 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } { 0 } / { x } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ \lim_{ x rarr infty } 0 #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ 0. #
# "This is our answer." #
#"Summarizing, we have shown:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \lim_{ x rarr infty } { 2 x^2 + 2 x + 3 } / { x^3 + x^2 + 1 } \ = \ 0. #
