Determine the following limit. \[ \lim _{x \rightarrow \infty} \frac{35 x^{3}+3 x^{2}-2 x}{14 x^{3}+x^{2}+5 x+4} \]

Step 1 ：We are given the limit \(\lim _{x \rightarrow \infty} \frac{35 x^{3}+3 x^{2}-2 x}{14 x^{3}+x^{2}+5 x+4}\).

Step 2 ：When x approaches infinity, the term with the highest degree in both the numerator and the denominator dominates the value of the function.

Step 3 ：We simplify the limit by dividing both the numerator and the denominator by \(x^{3}\), the highest degree in the function. This gives us a simpler limit to calculate.

Step 4 ：The simplified function is \(\frac{35+\frac{3}{x}-\frac{2}{x^{2}}}{14+\frac{1}{x}+\frac{5}{x^{2}}+\frac{4}{x^{3}}}\).

Step 5 ：As x approaches infinity, the terms \(\frac{3}{x}\), \(-\frac{2}{x^{2}}\), \(\frac{1}{x}\), \(\frac{5}{x^{2}}\), and \(\frac{4}{x^{3}}\) all approach 0.

Step 6 ：Therefore, the limit of the simplified function as x approaches infinity is \(\frac{35}{14}\).

Step 7 ：Final Answer: \(\boxed{\frac{35}{14}}\)