f) lim_(n rarr+oo)[(2)/(n)+((3)/(5))^(n)]

Analyze Sequence: Analyze the sequence. We have a sequence with two terms: (2/n) and (3/5)^n. We will take the limit of each term separately as n approaches infinity.Limit of (2/n): Take the limit of the first term (2/n) as n approaches infinity. The term (2/n) approaches 0 as n becomes very large, because the numerator is constant and the denominator grows without bound. lim_(n -> +oo) (2/n) = 0Limit of (3/5)^n: Take the limit of the second term (3/5)^n as n approaches infinity. The term (3/5)^n also approaches 0 as n becomes very large, because 3/5 is a fraction less than 1, and any fraction less than 1 raised to an increasingly large power will approach 0. lim_(n -> +oo) (3/5)^n = 0Combine Limits: Combine the limits of the two terms. Since both terms approach 0 as n approaches infinity, the limit of the entire sequence is the sum of the limits of the individual terms. lim_(n -> +oo) [(2/n) + (3/5)^n] = lim_(n -> +oo) (2/n) + lim_(n -> +oo) (3/5)^n = 0 + 0 = 0

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$f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]$

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f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]

Analyze Sequence: Analyze the sequence. $\newline$ We have a sequence with two terms: $(\frac{2}{n})$ and $(\frac{3}{5})^n$ . We will take the limit of each term separately as $n$ approaches infinity.
Limit of $(2/n)$ : Take the limit of the first term $(2/n)$ as $n$ approaches infinity. The term $(2/n)$ approaches $0$ as $n$ becomes very large, because the numerator is constant and the denominator grows without bound. $\lim_{n \to +\infty} (2/n) = 0$
Limit of $(\frac{3}{5})^n$ : Take the limit of the second term $(\frac{3}{5})^n$ as $n$ approaches infinity. $\newline$ The term $(\frac{3}{5})^n$ also approaches $0$ as $n$ becomes very large, because $\frac{3}{5}$ is a fraction less than $1$ , and any fraction less than $1$ raised to an increasingly large power will approach $0$ . $\newline$ $(\frac{3}{5})^n$ $0$
Combine Limits: Combine the limits of the two terms. $\newline$ Since both terms approach $0$ as $n$ approaches infinity, the limit of the entire sequence is the sum of the limits of the individual terms. $\newline$ \lim_{n \to +\infty} \left[\frac{$2$}{n} + \left(\frac{$3$}{$5$}\right)^n\right] = \lim_{n \to +\infty} \left(\frac{$2$}{n}\right) + \lim_{n \to +\infty} \left(\frac{$3$}{$5$}\right)^n = \(0 + $0$ = $0$