031 Review Part 1, Problem 8

True or false: Let  $W$  be a subspace of  $\mathbb {R} ^{4}$  and  ${\vec {v}}$  be a vector in  $\mathbb {R} ^{4}.$  If  ${\vec {v}}\in W$  and  ${\vec {v}}\in W^{\perp },$  then  ${\vec {v}}={\vec {0}}.$

Solution:
First, we switch to the limit to $x$ so that we can use L'Hopital's rule.
So, we have
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow \infty }{\frac {3-2x^{2}}{5x^{2}+x+1}}}&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow \infty }{\frac {-4x}{10x+1}}}\\&&\\&{\overset {L'H}{=}}&\displaystyle {\frac {-4}{10}}\\&&\\&=&\displaystyle {-{\frac {2}{5}}}.\end{array}}$

Final Answer:
False

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