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Limit of a Sequence

$\lim_{x\to C} f(x)=L$

Properties of Limits

Sum: $\lim_{x\to C} [f(x)+g(x)]=\lim_{x\to C} f(x)+\lim_{x\to C} g(x)$

Difference: $\lim_{x\to C} [f(x)-g(x)]=\lim_{x\to C} f(x)-\lim_{x\to C} g(x)$

Quotient: $\lim_{x\to C} \frac{f(x)}{g(x)}=\frac{\lim_{x\to C} f(x)}{\lim_{x\to C} g(x)}$

Product: $\lim_{x\to C} [f(x)\cdot g(x)]=\lim_{x\to C} f(x)\cdot\lim_{x\to C} g(x)$

Example: Find the limit as $n\rightarrow\infty$ of the sequence $\frac{2n-1}{3n+2}$

$\lim_{n\to\ \infty} (\frac{2n-1}{3n+2})=\lim_{n\to\ \infty} (\frac{2-\frac{1}{n}}{3+\frac{2}{n}})$
$=\frac{\lim_{n\to\ \infty} 2-\lim_{n\to\ \infty} \frac{1}{n}}{\lim_{n\to\ \infty} 3+\lim_{n\to\ \infty} \frac{2}{n}}\ \rightarrow \lim_{x\to\ \infty} 2\cdot\lim_{x\to\ \infty} \frac{1}{n}$
$=\frac{2-0}{3+2\cdot 0}$
$=\frac{2}{3}$