Explain how to find the value of each limit c) \( \lim _{x \rightarrow-\infty}-\frac{5 x^{5}+3 x^{3}+3}{6 x^{6}+2 x^{2}-9} \)

To evaluate   limₓ→₋∞ - (5x⁵ + 3x³ + 3) / (6x⁶ + 2x² − 9), follow these steps: 1. Identify the highest power of x in both the numerator and the denominator.   • In the numerator, the highest power is x⁵ (from 5x⁵).   • In the denominator, the highest power is x⁶ (from 6x⁶). 2. Notice that because the degree of the numerator (5) is less than the degree of the denominator (6), the fraction (before considering the negative sign) will tend to 0 as x tends to –∞. However, let’s show this more formally. 3. Divide both the numerator and the denominator by x⁶ (the highest power in the denominator):   Numerator: (5x⁵ + 3x³ + 3) / x⁶     = 5/x + 3/x³ + 3/x⁶   Denominator: (6x⁶ + 2x² − 9) / x⁶     = 6 + 2/x⁴ − 9/x⁶ 4. The expression for the limit becomes:   limₓ→₋∞ - [ (5/x + 3/x³ + 3/x⁶) / (6 + 2/x⁴ − 9/x⁶) ]. 5. As x tends to –∞, all terms with x in the denominator approach 0:   • In the numerator: 5/x → 0, 3/x³ → 0, 3/x⁶ → 0   • In the denominator: 2/x⁴ → 0, 9/x⁶ → 0   Thus, the expression simplifies to:   - [ 0 / 6 ] = 0. 6. Therefore, the value of the limit is 0. The key idea is that when the degree in the denominator is higher than in the numerator, the fraction goes to 0 as x goes to ±∞, and in this case, the extra negative sign doesn't affect that conclusion. Final answer: 0.