6.118 lim _(xarrow infty )((2x^3-3x^2+x+1)/(2x^3)-3x^(2-2x+3))^5x^(2)

To evaluate the limit $\lim _{x\rightarrow \infty }(\frac {2x^{3}-3x^{2}+x+1}{2x^{3}-3x^{2}-2x+3})^{5x^{2}}$, we can use the following steps:<br /><br />1. Divide both the numerator and denominator by $x^3$, the highest power of $x$ in the denominator:<br />$\lim _{x\rightarrow \infty }(\frac {2-\frac{3}{x}+\frac{1}{x^2}+\frac{1}{x^3}}{2-\frac{3}{x}-\frac{2}{x^2}+\frac{3}{x^3}})^{5x^{2}}$<br /><br />2. As $x$ approaches infinity, the terms with negative powers of $x$ will approach 0:<br />$\lim _{x\rightarrow \infty }(\frac {2}{2})^{5x^{2}}$<br /><br />3. Simplify the expression:<br />$\lim _{x\rightarrow \infty }(1)^{5x^{2}}$<br /><br />4. Since any number raised to the power of infinity is 1:<br />$\lim _{x\rightarrow \infty }(1)^{5x^{2}} = 1$<br /><br />Therefore, the correct answer is 1.