7. \( \lim _{x \rightarrow 8} \sqrt{\frac{10 x^{3}+2}{1+6 x^{3}}} \)

To find the limit \( \lim_{x \rightarrow 8} \sqrt{\frac{10x^3 + 2}{1 + 6x^3}} \), we can first evaluate the expression by directly substituting \( x = 8 \). Evaluating the numerator: \[ 10 \cdot 8^3 + 2 = 10 \cdot 512 + 2 = 5120 + 2 = 5122 \] Now evaluating the denominator: \[ 1 + 6 \cdot 8^3 = 1 + 6 \cdot 512 = 1 + 3072 = 3073 \] Now we can substitute these values back into the limit: \[ \lim_{x \rightarrow 8} \sqrt{\frac{10x^3 + 2}{1 + 6x^3}} = \sqrt{\frac{5122}{3073}} \] Now, we should calculate this value: \[ \sqrt{\frac{5122}{3073}} \approx 1.294 \] Thus, the limit is: \[ \lim_{x \rightarrow 8} \sqrt{\frac{10x^3 + 2}{1 + 6x^3}} \approx \sqrt{\frac{5122}{3073}} \]