find the following limit. if the limit is infinite, enter infinity or -infinity. if the limit is not…

Explanation:

Step1: Divide numerator and denominator by $e^{5x}$

$\lim_{x\rightarrow\infty}\frac{7 + 8e^{5x}}{6+9e^{5x}}=\lim_{x\rightarrow\infty}\frac{\frac{7}{e^{5x}}+8}{\frac{6}{e^{5x}} + 9}$

Step2: Evaluate the limit of each term

As $x\rightarrow\infty$, $\lim_{x\rightarrow\infty}\frac{7}{e^{5x}} = 0$ and $\lim_{x\rightarrow\infty}\frac{6}{e^{5x}}=0$ since the exponential - function $y = e^{ax}$ with $a>0$ grows much faster than a constant as $x\rightarrow\infty$. So, $\lim_{x\rightarrow\infty}\frac{\frac{7}{e^{5x}}+8}{\frac{6}{e^{5x}} + 9}=\frac{0 + 8}{0+9}$

Answer:

$\frac{8}{9}$