Question 3

Describe the long run behavior of $f(n)=5\left(\frac{1}{4}\right)^{n}+2$
As $n \rightarrow-\infty, f(n) \rightarrow ? \vee$
As $n \rightarrow \infty, f(n) \rightarrow ? \vee$

Step 1 ：The function \(f(n)=5\left(\frac{1}{4}\right)^{n}+2\) is an exponential function.

Step 2 ：As \(n\) approaches infinity, the term \(\left(\frac{1}{4}\right)^{n}\) will approach 0 because the base \(\frac{1}{4}\) is less than 1. Therefore, \(f(n)\) will approach 2.

Step 3 ：As \(n\) approaches negative infinity, the term \(\left(\frac{1}{4}\right)^{n}\) will approach infinity because the base \(\frac{1}{4}\) is less than 1 and the exponent is negative. Therefore, \(f(n)\) will approach infinity.

Step 4 ：However, it's not common to consider the behavior of exponential functions as the exponent approaches negative infinity, because in many real-world applications, the exponent represents time or some other quantity that can't be negative.

Step 5 ：Final Answer: As \(n \rightarrow-\infty, f(n) \rightarrow \infty \vee\)

Step 6 ：As \(n \rightarrow \infty, f(n) \rightarrow \boxed{2} \vee\)