8 mark for review lim x→∞ 10 - 6x² / 5 + 3e^x is a -2 b 0 c 2 d nonexistent

Explanation:

Step1: Analyze growth - rate of functions

As $x\to\infty$, the exponential function $e^{x}$ grows much faster than the polynomial function $x^{2}$.

Step2: Divide numerator and denominator by $e^{x}$

We have $\lim_{x\to\infty}\frac{10 - 6x^{2}}{5+3e^{x}}=\lim_{x\to\infty}\frac{\frac{10}{e^{x}}-\frac{6x^{2}}{e^{x}}}{\frac{5}{e^{x}} + 3}$.

Step3: Evaluate limits of individual terms

We know that $\lim_{x\to\infty}\frac{10}{e^{x}} = 0$, $\lim_{x\to\infty}\frac{5}{e^{x}}=0$, and using L'Hopital's rule twice on $\lim_{x\to\infty}\frac{6x^{2}}{e^{x}}$: First - application: $\lim_{x\to\infty}\frac{6x^{2}}{e^{x}}=\lim_{x\to\infty}\frac{12x}{e^{x}}$. Second - application: $\lim_{x\to\infty}\frac{12x}{e^{x}}=\lim_{x\to\infty}\frac{12}{e^{x}} = 0$. So, $\lim_{x\to\infty}\frac{\frac{10}{e^{x}}-\frac{6x^{2}}{e^{x}}}{\frac{5}{e^{x}} + 3}=\frac{0 - 0}{0 + 3}=0$.

Answer:

B. 0