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}