Limits of sequences defined using continuous functions

## Limits of sequences defined using continuous functions
            ---
            ## Continuity and limits of sequences
            #### Theorem
            Let $I$ be an open interval, $c\in I$, $f$ defined on $I$ except possibly at $c$, and $\displaystyle\lim\_{x\to c}f(x)=\ell$.
            Suppose $(a_n)\_{n=1}^\infty$ be a sequence with
            1. all $a_n\in I$,
            1. all $a_n\neq c$ or $f$ continuous at $c$, and
            1. $\displaystyle\lim\_{n\to\infty}a_n=c$.

Then $\displaystyle\lim\_{n\to\infty}f(a_n)=\ell$.
            
            ---
            ## Continuity and limits of sequences
            #### Theorem
            Let $I$ be an open interval, $c\in I$, $f$ defined on $I$ except possibly at $c$, and $\displaystyle\lim\_{x\to c}f(x)=\ell$.
            Suppose $(a_n)\_{n=1}^\infty$ be a sequence with
            1. all $a_n\in I$,
            1. all $a_n\neq c$ or $f$ continuous at $c$, and
            1. $\displaystyle\lim\_{n\to\infty}a_n=c$.

Then $\displaystyle\lim\_{n\to\infty}f(a_n)=\ell$.
            #### Example
            $\displaystyle \lim\_{n\to\infty}\sqrt[5]{7+\frac{3n^2-1}{5-n^2}} = \sqrt[5]{\lim\_{n\to\infty}\left[ 7+\frac{3n^2-1}{5-n^2} \right]} = \sqrt[5]{7-3} = \sqrt[5]{4}$.
            In the first equality, we used continuity of the fifth root function on $\mathbb R^+$ and $7+\frac{3n^2-1}{5-n^2}>0$ for all $n\in\mathbb N$.
            
            ---
            ## Continuity and limits of sequences
            #### Theorem
            Let $I$ be an open interval, $c\in I$, $f$ defined on $I$ except possibly at $c$, and $\displaystyle\lim\_{x\to c}f(x)=\ell$.
            Suppose $(a_n)\_{n=1}^\infty$ be a sequence with
            1. all $a_n\in I$,
            1. all $a_n\neq c$ or $f$ continuous at $c$, and
            1. $\displaystyle\lim\_{n\to\infty}a_n=c$.

Then $\displaystyle\lim\_{n\to\infty}f(a_n)=\ell$.
            #### Example
            $\displaystyle \lim\_{n\to\infty}\left\lvert \frac{3n+2}{7n-103} \right\rvert={}$?

$\displaystyle \lim\_{n\to\infty}\cos\left( \pi + \frac{4}{n^2}\sin(n) \right)={}$?
            
            ---
            ## Continuity and limits of sequences
            #### Theorem
            Let $I$ be an open interval, $c\in I$, $f$ defined on $I$ except possibly at $c$, and $\displaystyle\lim\_{x\to c}f(x)=\ell$.
            Suppose $(a_n)\_{n=1}^\infty$ be a sequence with
            1. all $a_n\in I$,
            1. all $a_n\neq c$ or $f$ continuous at $c$, and
            1. $\displaystyle\lim\_{n\to\infty}a_n=c$.

Then $\displaystyle\lim\_{n\to\infty}f(a_n)=\ell$.
            #### Example
            $\displaystyle \lim\_{n\to\infty}\left\lvert \frac{3n+2}{7n-103} \right\rvert = \left\lvert \lim\_{n\to\infty} \frac{3n+2}{7n-103} \right\rvert = \left\lvert \frac37 \right\rvert$, because the absolute value function is continuous on $\mathbb R$, and $7n-103\neq0$ for any natural number $n$.
            
            ---
            ## Continuity and limits of sequences
            #### Theorem
            Let $I$ be an open interval, $c\in I$, $f$ defined on $I$ except possibly at $c$, and $\displaystyle\lim\_{x\to c}f(x)=\ell$.
            Suppose $(a_n)\_{n=1}^\infty$ be a sequence with
            1. all $a_n\in I$,
            1. all $a_n\neq c$ or $f$ continuous at $c$, and
            1. $\displaystyle\lim\_{n\to\infty}a_n=c$.

Then $\displaystyle\lim\_{n\to\infty}f(a_n)=\ell$.
            #### Example
            $\displaystyle \lim\_{n\to\infty}\cos\left( \pi + \frac{4}{n^2}\sin(n) \right) = \cos\left( \lim\_{n\to\infty}\left[ \pi + \frac{4}{n^2}\sin(n) \right] \right) = \cos\left( \pi + \lim\_{n\to\infty}\left[ \frac{4}{n^2}\sin(n) \right] \right) = \cos(\pi+0)=-1$, using continuity of cosine on $\mathbb R$, and we can use the squeeze theorem to evaluate the final limit.