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\begin{aligned} \lim\limits_{x \rightarrow 2} \frac{2 x^{2}-3 x+1}{x^{3}+4} &=\frac{\lim\limits_{x \rightarrow 2}\left(2 x^{2}-3 x+1\right)}{\lim\limits_{x \rightarrow 2}\left(x^{3}+4\right)} \qquad \qquad \text { Apply the quotient law, making sure that. }(2)^{3}+4 \neq 0\ \\ &=\frac{2 \cdot \lim\limits_{x \rightarrow 2} x^{2}-3 \cdot \lim\limits_{x \rightarrow 2} x+\lim\limits_{x \rightarrow 2}}{\lim\limits_{x \rightarrow 2} x^{3}+\lim\limits_{x \rightarrow 2} 4} \qquad \text{Apply the sum law and constant multiple law.} \\ &=\frac{2 \cdot\left(\lim\limits_{x \rightarrow 2} x\right)^{2}-3 \cdot \lim\limits_{x \rightarrow 2} x+\lim\limits_{x \rightarrow 2}}{\left(\lim\limits_{x \rightarrow 2} x\right)^{3}+\lim\limits_{x \rightarrow 2} 4} \qquad \text{Apply the power law.} \\ &=\frac{2(4)-3(2)+1}{(2)^{3}+4}=\frac{1}{4} \qquad \qquad \text{Apply the basic limit laws and simplify.} \end{aligned}