Answer
See below.
Work Step by Step
Theorem 1 is used when the terms of a sequence can be expressed as a sum, difference, product or quotient, or a constant multiple of terms of known convergent sequences.
For example $a_{n}=\displaystyle \frac{1}{2n}+\frac{1}{n}$
Since $\displaystyle \{\frac{1}{n}\}$ and $\displaystyle \{\frac{1}{2n}\} $ both converge to 0, then:
$\{a_{n}\}$ converges to $0+0=0$
Theorem 2, known as The Sandwich Theorem for Sequences,
is used if we can find two convergent sequences that converge to the same number L, such that we can envelop each term of our sequence between the terms of the two.
Example:
$\{0\}$ and $\displaystyle \{\frac{1}{n}\}$ both converge to $L=0.$
Since $0\displaystyle \leq\frac{1}{n^{2}}\leq\frac{1}{n}$, it follows that $\displaystyle \{\frac{1}{n^{2}}\}$ also converges to $0$.
Theorem 3, or The Continuous Function Theorem for Sequences
allows for a continuous function $f$:
If $a_{n}\rightarrow L$, then $f(a_{n})\rightarrow f(L)$.
Example:
$\displaystyle \lim_{n\rightarrow\infty}\ln(1+\frac{1}{n})=\ln[\lim_{n\rightarrow\infty}(1+\frac{1}{n})]=\ln 1=0$
Theorem 5 lists 6 standard convergent sequences that are used in previously mentioned methods.
