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Determine the limit of the sequence.

[tex]\[ a_n = e^{\frac{2n}{3n+6}} \][/tex]

(Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.)

[tex]\[ \lim_{n \rightarrow \infty} a_n = \boxed{\ } \][/tex]

Sagot :

To determine the limit of the sequence given by [tex]\( a_n = e^{\frac{2n}{3n + 6}} \)[/tex] as [tex]\( n \)[/tex] approaches infinity, follow these steps:

1. Analyze the exponent: We first focus on the expression inside the exponent, which is [tex]\( \frac{2n}{3n + 6} \)[/tex].

2. Simplify the fraction: To simplify [tex]\( \frac{2n}{3n + 6} \)[/tex], divide both the numerator and the denominator by [tex]\( n \)[/tex]:
[tex]\[ \frac{2n}{3n + 6} = \frac{2n / n}{(3n + 6) / n} = \frac{2}{3 + \frac{6}{n}} \][/tex]

3. Evaluate the limit inside the exponent: As [tex]\( n \)[/tex] approaches infinity, the term [tex]\( \frac{6}{n} \)[/tex] approaches 0. Therefore, the expression [tex]\( 3 + \frac{6}{n} \)[/tex] approaches 3, and so:
[tex]\[ \lim_{n \to \infty} \frac{2}{3 + \frac{6}{n}} = \frac{2}{3} \][/tex]

4. Apply the limit to the entire expression: Now we evaluate the limit of the entire expression [tex]\( a_n \)[/tex]. Since the exponent [tex]\( \frac{2n}{3n + 6} \)[/tex] approaches [tex]\( \frac{2}{3} \)[/tex] as [tex]\( n \)[/tex] approaches infinity, we get:
[tex]\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} e^{\frac{2n}{3n + 6}} = e^{\frac{2}{3}} \][/tex]

Therefore, the limit of the sequence [tex]\( a_n = e^{\frac{2n}{3n + 6}} \)[/tex] as [tex]\( n \)[/tex] approaches infinity is:

[tex]\[ \lim_{n \to \infty} a_n = e^{\frac{2}{3}} \][/tex]